A rectangle has a length that is times its width. The function represents this situation, where is the area, in square feet, of the rectangle and . Which of the following is the best interpretation of in this context?
The length of the rectangle, in feet
The area of the rectangle, in square feet
The difference between the length and the width of the rectangle, in feet
The width of the rectangle, in feet
Choice A is correct. It's given that a rectangle has a length that is times its width. It's also given that the function represents this situation, where is the area, in square feet, of the rectangle and . The area of a rectangle can be calculated by multiplying the rectangle's length by its width. Since the rectangle has a length that is times its width, it follows that represents the width of the rectangle, in feet, and represents the length of the rectangle, in feet. Therefore, the best interpretation of in this context is that it's the length of the rectangle, in feet.
Choice B is incorrect. This is the best interpretation of , not , in the given function.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect. This is the best interpretation of , not , in the given function.
The given equation relates the distinct positive numbers , , and . Which equation correctly expresses in terms of and ?
Choice A is correct. It’s given that is positive. Therefore, multiplying each side of the given equation by yields , which is equivalent to . Thus, the equation correctly expresses in terms of and .
Choice B is incorrect. This equation is equivalent to .
Choice C is incorrect. This equation is equivalent to .
Choice D is incorrect. This equation is equivalent to .
Which expression is equivalent to , where , , and are positive?
Choice B is correct. Applying the commutative property of multiplication, the expression can be rewritten as . For positive values of , . Therefore, the expression can be rewritten as , or .
Choice A is incorrect and may result from multiplying, not adding, the exponents.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
A system of equations consists of a quadratic equation and a linear equation. The equations in this system are graphed in the xy-plane above. How many solutions does this system have?
0
1
2
3
Choice C is correct. The solutions to a system of two equations correspond to points where the graphs of the equations intersect. The given graphs intersect at 2 points; therefore, the system has 2 solutions.
Choice A is incorrect because the graphs intersect. Choice B is incorrect because the graphs intersect more than once. Choice D is incorrect. It’s not possible for the graph of a quadratic equation and the graph of a linear equation to intersect more than twice.
Which of the following inequalities is equivalent to the inequality above?
Choice B is correct. Both sides of the given inequality can be divided by 3 to yield .
Choices A, C, and D are incorrect because they are not equivalent to (do not have the same solution set as) the given inequality. For example, the ordered pair is a solution to the given inequality, but it is not a solution to any of the inequalities in choices A, C, or D.
If is a solution to the system of equations above, which of the following could be the value of x ?
–1
0
2
3
Choice A is correct. It is given that y = x + 1 and y = x2 + x. Setting the values for y equal to each other yields x + 1 = x2 + x. Subtracting x from each side of this equation yields x2 = 1. Therefore, x can equal 1 or –1. Of these, only –1 is given as a choice.
Choice B is incorrect. If x = 0, then x + 1 = 1, but x2 + x = 02 + 0 = 0 ≠︀ 1. Choice C is incorrect. If x = 2, then x + 1 = 3, but x2 + x = 22 + 2 = 6 ≠︀ 3. Choice D is incorrect. If x = 3, then x + 1 = 4, but x2 + x = 32 + 3 = 12 ≠︀ 4.
Which expression is equivalent to ?
Choice B is correct. Since is a common factor of each term in the given expression, the expression can be rewritten as .
Choice A is incorrect. This expression is equivalent to .
Choice C is incorrect. This expression is equivalent to .
Choice D is incorrect. This expression is equivalent to .
Which of the following is a factor of the polynomial above?
Choice B is correct. The first and last terms of the polynomial are both squares such that and
. The second term is twice the product of the square root of the first and last terms:
. Therefore, the polynomial is the square of a binomial such that
, and
is a factor.
Choice A is incorrect and may be the result of incorrectly factoring the polynomial. Choice C is incorrect and may be the result of dividing the second and third terms of the polynomial by their greatest common factor. Choice D is incorrect and may be the result of not factoring the coefficients.
If and
, which of the following is equivalent to
?
Choice B is correct. It’s given that and
. Substituting the values for p and v into the expression
yields
. Multiplying the terms
yields
. Using the distributive property to rewrite
yields
. Therefore, the entire expression can be represented as
. Combining like terms yields
.
Choice A is incorrect and may result from subtracting, instead of adding, the term . Choice C is incorrect. This is the result of multiplying the terms
. Choice D is incorrect and may result from distributing 2, instead of
, to the term
.
What values satisfy the equation above?
and
and
and
and
Choice C is correct. Using the quadratic formula to solve the given expression yields . Therefore,
and
satisfy the given equation.
Choices A and B are incorrect and may result from incorrectly factoring or incorrectly applying the quadratic formula. Choice D is incorrect and may result from a sign error.
How many distinct real solutions are there to the given equation?
Exactly one
Exactly two
Infinitely many
Zero
Choice B is correct. The number of solutions to a quadratic equation of the form , where , , and are constants, can be determined by the value of the discriminant, . If the value of the discriminant is positive, then the quadratic equation has exactly two distinct real solutions. If the value of the discriminant is equal to zero, then the quadratic equation has exactly one real solution. If the value of the discriminant is negative, then the quadratic equation has zero real solutions. In the given equation, , , , and . Substituting for , for , and for in yields , or . Since the value of the discriminant is positive, the given equation has exactly two distinct real solutions.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The product of two positive integers is . If the first integer is greater than twice the second integer, what is the smaller of the two integers?
The correct answer is . Let represent the first integer and represent the second integer. If the first integer is greater than twice the second integer, then . It's given that the product of the two integers is ; therefore . Substituting for in this equation yields , which can be written as . Subtracting from each side of this equation yields . The left-hand side of this equation can be factored by finding two values whose product is , or , and whose sum is . The two values whose product is and whose sum is are and . Thus, the equation can be rewritten as , which is equivalent to , or . By the zero product property, it follows that or . Subtracting from both sides of the equation yields . Dividing both sides of this equation by yields . Since is a positive integer, the value of isn't . Adding to both sides of the equation yields . Substituting for in the equation yields . Dividing both sides of this equation by yields . Therefore, the two integers are and , so the smaller of the two integers is .
The function is defined by , where . What is the product of and ?
The correct answer is . It’s given that the function is defined by , where . Substituting for in function yields . This function can be rewritten as , or . Since , it follows that . Substituting for in yields , or . Similarly, substituting for in function yields . This function can be rewritten as , or . Since , it again follows that . Substituting for in yields , or . Therefore, and . Thus, the product of and is , or .
The graph shown gives the estimated value, in dollars, of a tablet as a function of the number of months since it was purchased. What is the best interpretation of the y-intercept of the graph in this context?
The estimated value of the tablet was when it was purchased.
The estimated value of the tablet months after it was purchased was .
The estimated value of the tablet had decreased by in the months after it was purchased.
The estimated value of the tablet decreased by approximately each year after it was purchased.
Choice A is correct. It's given that the graph shown gives the estimated value , in dollars, of a tablet as a function of the number of months since it was purchased, . The y-intercept of a graph is the point at which the graph intersects the y-axis, or when is . The graph shown intersects the y-axis at the point . It follows that months after the tablet was purchased, or when the tablet was purchased, the estimated value of the tablet was dollars. Therefore, the best interpretation of the y-intercept is that the estimated value of the tablet was when it was purchased.
Choice B is incorrect. The estimated value of the tablet months after it was purchased was , not .
Choice C is incorrect. The estimated value of the tablet had decreased by , or , not , in the months after it was purchased.
Choice D is incorrect and may result from conceptual errors.
Which of the following is equivalent to the given expression?
Choice B is correct. Using the associative and commutative properties of addition, the given expression can be rewritten as
. Adding these like terms results in
.
Choice A is incorrect and may result from adding . Choice C is incorrect and may result from adding
. Choice D is incorrect and may result from adding
.
The table shows three values of and their corresponding values of , where and is a linear function. What is the y-intercept of the graph of in the xy-plane?
Choice A is correct. It's given that the table shows values of and their corresponding values of , where . It's also given that is a linear function. It follows that an equation that defines can be written in the form , where represents the slope and represents the y-coordinate of the y-intercept of the graph of in the xy-plane. The slope of the graph of can be found using two points, and , that are on the graph of , and the formula . Since the table shows values of and their corresponding values of , substituting values of and in the equation can be used to define function . Using the first pair of values from the table, and , yields , or . Multiplying each side of this equation by yields , so the point is on the graph of . Using the second pair of values from the table, and , yields , or . Multiplying each side of this equation by yields , so the point is on the graph of . Substituting and for and , respectively, in the formula yields , or . Substituting for in the equation yields . Since , substituting for and for in the equation yields , or . Adding to both sides of this equation yields . It follows that is the y-coordinate of the y-intercept of the graph of . Therefore, the y-intercept of the graph of is .
Choice B is incorrect. is the y-coordinate of the y-intercept of the graph of .
Choice C is incorrect. is the slope of the graph of .
Choice D is incorrect. is the x-coordinate of the x-intercept of the graph of .
Which expression is equivalent to , where ?
Choice B is correct. The given expression has a common factor of in the denominator, so the expression can be rewritten as . The three terms in this expression have a common factor of . Since it's given that , can't be equal to , which means can't be equal to . Therefore, each term in the expression, , can be divided by , which gives .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The graphs of the given equations intersect at the point in the xy-plane. What is the value of ?
Choice A is correct. It's given that the graphs of the given equations intersect at the point in the xy-plane. It follows that represents a solution to the system consisting of the given equations. The first equation given is . Substituting for in the second equation given, , yields , which is equivalent to , or . It follows that the graphs of the given equations intersect at the point . Therefore, the value of is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The quadratic function h is defined as shown. In the xy-plane, the graph of intersects the x-axis at the points
and
, where t is a constant. What is the value of t ?
1
2
4
8
Choice D is correct. It’s given that the graph of intersects the x-axis at
and
, where t is a constant. Since this graph intersects the x-axis when
or when
, it follows that
and
. If
, then
. Adding 32 to both sides of this equation yields
. Dividing both sides of this equation by 2 yields
. Taking the square root of both sides of this equation yields
. Adding 4 to both sides of this equation yields
. Therefore, the value of t is 8.
Choices A, B, and C are incorrect and may result from calculation errors.
The graph shown will be translated up units. Which of the following will be the resulting graph?
Choice A is correct. When a graph is translated up units, each point on the resulting graph is units above the point on the original graph. In other words, the y-value of each point on the graph increases by . The graph shown passes through the points , , and . It follows that when the graph shown is translated up units, the resulting graph will pass through the points , , and . These points are , , and , respectively. Of the given choices, only the graph in choice A passes through the points , , and .
Choice B is incorrect. This is the result of translating the graph down, rather than up, units.
Choice C is incorrect. This is the result of translating the graph left, rather than up, units.
Choice D is incorrect. This is the result of translating the graph right, rather than up, units.
What is the minimum value of the given function?
The correct answer is . The given function can be rewritten in the form , where is a positive constant and the minimum value, , of the function occurs when the value of is . By completing the square, can be written as , or . This equation is in the form , where , , and . Therefore, the minimum value of the given function is .
What is the positive solution to the given equation?
The correct answer is . The given equation can be rewritten as . Dividing each side of this equation by yields . By the definition of absolute value, if , then or . Subtracting from each side of the equation yields . Dividing each side of this equation by yields . Similarly, subtracting from each side of the equation yields . Dividing each side of this equation by yields . Therefore, since the two solutions to the given equation are and , the positive solution to the given equation is .
The function is defined by . What is the y-intercept of the graph of in the xy-plane?
Choice A is correct. The y-intercept of the graph of in the xy-plane occurs at the point on the graph where . In other words, when , the corresponding value of is the y-coordinate of the y-intercept. Substituting for in the given equation yields , which is equivalent to , or . Thus, when , the corresponding value of is . Therefore, the y-intercept of the graph of in the xy-plane is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This could be the y-intercept for , not .
The graph shows the height above ground, in meters, of a ball seconds after the ball was launched upward from a platform. Which statement is the best interpretation of the marked point in this context?
second after being launched, the ball's height above ground is meters.
seconds after being launched, the ball's height above ground is meter.
The ball was launched from an initial height of meter with an initial velocity of meters per second.
The ball was launched from an initial height of meters with an initial velocity of meter per second.
Choice A is correct. It's given that the graph shows the height above ground, in meters, of a ball seconds after the ball was launched upward from a platform. In the graph shown, the x-axis represents time, in seconds, and the y-axis represents the height of the ball above ground, in meters. It follows that for the marked point , represents the time, in seconds, after the ball was launched upward from a platform and represents the height of the ball above ground, in meters. Therefore, the best interpretation of the marked point is second after being launched, the ball's height above ground is meters.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
An auditorium has seats for people. Tickets to attend a show at the auditorium currently cost . For each increase to the ticket price, fewer tickets will be sold. This situation can be modeled by the equation , where represents the increase in ticket price, in dollars, and represents the revenue, in dollars, from ticket sales. If this equation is graphed in the xy-plane, at what value of is the maximum of the graph?
Choice B is correct. It’s given that the situation can be modeled by the equation , where represents the increase in ticket price, in dollars, and represents the revenue, in dollars, from ticket sales. Since the coefficient of the term is negative, the graph of this equation in the xy-plane opens downward and reaches its maximum value at its vertex. If a quadratic equation in the form , where , , and are constants, is graphed in the xy-plane, the x-coordinate of the vertex is equal to . For the equation , , , and . It follows that the x-coordinate of the vertex is , or . Therefore, if the given equation is graphed in the xy-plane, the maximum of the graph occurs at an x-value of .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
One solution to the given equation can be written as , where is a constant. What is the value of ?
Choice B is correct. Adding to each side of the given equation yields . To complete the square, adding to each side of this equation yields , or . Taking the square root of each side of this equation yields . Adding to each side of this equation yields . Since it's given that one of the solutions to the equation can be written as , the value of must be .
Alternate approach: It's given that is a solution to the given equation. It follows that . Substituting for in the given equation yields , or . Expanding the products on the left-hand side of this equation yields , or . Adding to each side of this equation yields .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The given equation relates the value of and its corresponding value of for the function . What is the minimum value of the function ?
Choice B is correct. For a quadratic function defined by an equation of the form , where , , and are constants and , the minimum value of the function is . Subtracting from both sides of the given equation yields . This function is in the form , where , , and . Therefore, the minimum value of the function is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
In the -plane, a line with equation intersects a parabola at exactly one point. If the parabola has equation , where is a positive constant, what is the value of ?
The correct answer is . It’s given that a line with equation intersects a parabola with equation , where is a positive constant, at exactly one point in the xy-plane. It follows that the system of equations consisting of and has exactly one solution. Dividing both sides of the equation of the line by yields . Substituting for in the equation of the parabola yields . Adding and subtracting from both sides of this equation yields . A quadratic equation in the form of , where , , and are constants, has exactly one solution when the discriminant, , is equal to zero. Substituting for and for in the expression and setting this expression equal to yields , or . Adding to each side of this equation yields . Taking the square root of each side of this equation yields . It’s given that is positive, so the value of is .
The given function models the number of advertisements a company sent to its clients each year, where represents the number of years since , and . If is graphed in the xy-plane, which of the following is the best interpretation of the y-intercept of the graph in this context?
The minimum estimated number of advertisements the company sent to its clients during the years was .
The minimum estimated number of advertisements the company sent to its clients during the years was .
The estimated number of advertisements the company sent to its clients in was .
The estimated number of advertisements the company sent to its clients in was .
Choice D is correct. The y-intercept of a graph in the xy-plane is the point where . For the given function , the y-intercept of the graph of in the xy-plane can be found by substituting for in the equation , which gives . This is equivalent to , or . Therefore, the y-intercept of the graph of is . It’s given that the function models the number of advertisements a company sent to its clients each year. Therefore, represents the estimated number of advertisements the company sent to its clients each year. It's also given that represents the number of years since . Therefore, represents years since , or . Thus, the best interpretation of the y-intercept of the graph of is that the estimated number of advertisements the company sent to its clients in was .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Which expression is equivalent to ?
Choice D is correct. The given expression can be rewritten as . Combining like terms in this expression yields .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The first term of a sequence is . Each term after the first is times the preceding term. If represents the th term of the sequence, which equation gives in terms of ?
Choice D is correct. Since represents the th term of the sequence and is the first term of the sequence, the value of is when the value of is . Since each term after the first is times the preceding term, the value of is when the value of is . Therefore, the value of is , or , when the value of is . More generally, the value of is for a given value of . Therefore, the equation gives in terms of .
Choice A is incorrect. This equation describes a sequence for which the first term is , rather than , and each term after the first is , rather than , times the preceding term.
Choice B is incorrect. This equation describes a sequence for which the first term is , rather than , and each term after the first is , rather than , times the preceding term.
Choice C is incorrect. This equation describes a sequence for which the first term is , rather than .
The function is defined by , where and are integer constants and . The functions and are equivalent to function , where and are constants. Which of the following equations displays the y-coordinate of the y-intercept of the graph of in the xy-plane as a constant or coefficient?
I only
II only
I and II
Neither I nor II
Choice D is correct. A y-intercept of a graph in the xy-plane is a point where the graph intersects the y-axis, or a point where . Substituting for in the equation defining function yields , or . So, the y-coordinate of the y-intercept of the graph is , or equivalently, . It's given that function is equivalent to function , where . It follows that . Since can't be equal to , the coefficient can't be equal to . Since , the constant , which is equal to , can't be equal to . Therefore, function doesn't display the y-coordinate of the y-intercept of the graph of in the xy-plane as a constant or coefficient. It's also given that function is equivalent to function , where . The equation defining can be rewritten as . It follows that . Since can't be equal to , the coefficient can't be equal to . Since , the constant , which is equal to , can't be equal to . Therefore, function doesn't display the y-coordinate of the y-intercept of the graph of in the xy-plane as a constant or coefficient. Thus, neither function nor function displays the y-coordinate of the y-intercept of the graph of in the xy-plane as a constant or coefficient.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
A physics class is planning an experiment about a toy rocket. The equation gives the estimated height , in feet, of the toy rocket seconds after it is launched into the air. Which of the following is the best interpretation of the vertex of the graph of the equation in the xy-plane?
This toy rocket reaches an estimated maximum height of feet seconds after it is launched into the air.
This toy rocket reaches an estimated maximum height of feet seconds after it is launched into the air.
This toy rocket reaches an estimated maximum height of feet seconds after it is launched into the air.
This toy rocket reaches an estimated maximum height of feet seconds after it is launched into the air.
Choice B is correct. The vertex of the graph of a quadratic equation is where it reaches its minimum or maximum value. When a quadratic equation is written in the form , the vertex of the parabola represented by the equation is . In the given equation , the value of is and the value of is . It follows that the vertex of the graph of this equation in the xy-plane is . Additionally, since in the given equation, the graph of the quadratic equation opens down, and the vertex represents a maximum. It’s given that the value of represents the estimated height, in feet, of the toy rocket seconds after it is launched into the air. Therefore, this toy rocket reaches an estimated maximum height of feet seconds after it is launched into the air.
Choice A is incorrect. The in the equation is an indicator of how narrow the graph of the equation is rather than where it reaches its maximum.
Choice C is incorrect. The in the equation is an indicator of how narrow the graph of the equation is rather than where it reaches its maximum.
Choice D is incorrect. This is an interpretation of the vertex of the graph of the equation , not of the equation .
Which ordered pair is a solution to the system of equations above?
Choice A is correct. The solution to the given system of equations can be found by solving the first equation for x, which gives , and substituting that value of x into the second equation which gives
. Rewriting this equation by adding like terms and expanding
gives
. Subtracting
from both sides of this equation gives
. Adding to 2 to both sides of this equation gives
. Therefore, it follows that
. Substituting
for y in the first equation yields
. Adding
to both sides of this equation yields
. Therefore, the ordered pair
is a solution to the given system of equations.
Choice B is incorrect. Substituting for x and
for y in the first equation yields
, or
, which isn’t a true statement. Choice C is incorrect. Substituting
for x and
for y in the second equation yields
, or
, which isn’t a true statement. Choice D is incorrect. Substituting
for x and
for y in the second equation yields
, or
, which isn’t a true statement.
What is the -intercept of the graph shown?
Choice B is correct. The y-intercept of a graph in the xy-plane is the point on the graph where . At , the corresponding value of is . Therefore, the y-intercept of the graph shown is .
Choice A is incorrect and may result from conceptual errors.
Choice C is incorrect. This is the y-intercept of a graph in the xy-plane that intersects the y-axis at , not .
Choice D is incorrect. This is the y-intercept of a graph in the xy-plane that intersects the y-axis at , not .
Which of the following is equivalent to the expression ?
Choice B is correct. The term x4 can be factored as . Factoring –6 as
yields values that add to –1, the coefficient of x2 in the expression.
Choices A, C, and D are incorrect and may result from finding factors of –6 that don’t add to the coefficient of x2 in the original expression.
Which of the following is equivalent to the expression above?
Choice A is correct. The given expression can be rewritten as . Applying the distributive property, the expression
can be rewritten as
, or
. Adding like terms yields
. Substituting
for
in the given expression yields
. By the rules of exponents, the expression
is equivalent to
. Applying the distributive property, this expression can be rewritten as
, or
. Adding like terms gives
. Substituting
for
in the rewritten expression yields
, and adding like terms yields
.
Choices B, C, and D are incorrect. Choices C and D may result from rewriting the expression as
, instead of as
. Choices B and C may result from rewriting the expression
as
, instead of
.
| Time (years) | Total amount (dollars) |
|---|---|
Rosa opened a savings account at a bank. The table shows the exponential relationship between the time , in years, since Rosa opened the account and the total amount , in dollars, in the account. If Rosa made no additional deposits or withdrawals, which of the following equations best represents the relationship between and ?
Choice C is correct. It’s given that the relationship between and is exponential. The table shows that the value of increases as the value of increases. Therefore, the relationship between and can be represented by an increasing exponential equation of the form , where and are positive constants. The table shows that when , . Substituting for and for in the equation yields , which is equivalent to , or . Substituting for in the equation yields . The table also shows that when , . Substituting for and for in the equation yields , or . Dividing both sides of this equation by yields approximately . Subtracting from both sides of this equation yields that the value of is approximately . Substituting for in the equation yields . Therefore, of the choices, choice C best represents the relationship between and .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The equation above is true for all x, where a and b are constants. What is the value of ab ?
18
20
24
40
Choice C is correct. If the equation is true for all x, then the expressions on both sides of the equation will be equivalent. Multiplying the polynomials on the left-hand side of the equation gives . On the right-hand side of the equation, the only
-term is
. Since the expressions on both sides of the equation are equivalent, it follows that
, which can be rewritten as
. Therefore,
, which gives
.
Choice A is incorrect. If , then the coefficient of
on the left-hand side of the equation would be
, which doesn’t equal the coefficient of
,
, on the right-hand side. Choice B is incorrect. If
, then the coefficient of
on the left-hand side of the equation would be
, which doesn’t equal the coefficient of
,
, on the right-hand side. Choice D is incorrect. If
, then the coefficient of
on the left-hand side of the equation would be
, which doesn’t equal the coefficient of
,
, on the right-hand side.
How many distinct real solutions does the given equation have?
Exactly two
Exactly one
Zero
Infinitely many
Choice A is correct. The number of solutions of a quadratic equation of the form , where , , and are constants, can be determined by the value of the discriminant, . If the value of the discriminant is positive, then the quadratic equation has exactly two distinct real solutions. If the value of the discriminant is equal to zero, then the quadratic equation has exactly one real solution. If the value of the discriminant is negative, then the quadratic equation has zero real solutions. In the given equation, , , , and . Substituting these values for , , and in yields , or . Since the value of its discriminant is positive, the given equation has exactly two distinct real solutions.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
For the exponential function , the table shows four values of and their corresponding values of . Which equation defines ?
Choice D is correct. It's given that function is exponential. Therefore, an equation defining can be written in the form , where and are constants. The table shows that when , . Substituting for and for in the equation yields , which is equivalent to . Substituting for in the equation yields . The table also shows that when , . Substituting for and for in the equation yields , which is equivalent to . Substituting for in the equation yields .
Choice A is incorrect. For this function, is equal to , not .
Choice B is incorrect. For this function, is equal to , not .
Choice C is incorrect. For this function, is equal to , not .
Which of the following expressions is equivalent to ?
Choice C is correct. The expression can be written as a difference of squares x2 – y2, which can be factored as (x + y)(x – y). Here, y2 = 5, so , and the expression therefore factors as
.
Choices A and B are incorrect and may result from misunderstanding how to factor a difference of squares. Choice D is incorrect; (x + 5)(x – 1) can be rewritten as x2 + 4x – 5, which is not equivalent to the original expression.
Which of the following expressions is(are) a factor of ?
I only
II only
I and II
Neither I nor II
Choice B is correct. The given expression can be factored by first finding two values whose sum is and whose product is , or . Those two values are and . It follows that the given expression can be rewritten as . Since the first two terms of this expression have a common factor of and the last two terms of this expression have a common factor of , this expression can be rewritten as . Since the two terms of this expression have a common factor of , it can be rewritten as . Therefore, expression II, , is a factor of , but expression I, , is not a factor of .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
If for all positive values of x, what is the value of
?
The correct answer is . The value of
can be found by first rewriting the left-hand side of the given equation as
. Using the properties of exponents, this expression can be rewritten as
. This expression can be rewritten by subtracting the fractions in the exponent, which yields
. Thus,
is
. Note that 7/6, 1.166, and 1.167 are examples of ways to enter a correct answer.
The table shows the exponential relationship between the number of years, , since Hana started training in pole vault, and the estimated height , in meters, of her best pole vault for that year. Which of the following functions best represents this relationship, where ?
Choice D is correct. The table shows an increasing exponential relationship between the number of years, , since Hana started training in pole vault and the estimated height , in meters, of her best pole vault for that year. The relationship can be written as , where and are positive constants. It's given that when , . Substituting for and for in yields , or . Substituting for in yields . It's also given that when , . Substituting for and for in yields . Dividing each side of this equation by yields , or is approximately equal to . Since is positive, is approximately equal to , or . Substituting for in yields .
Choice A is incorrect. When , the value of in this function is equal to rather than , and it is decreasing rather than increasing.
Choice B is incorrect. When , the value of in this function is equal to rather than .
Choice C is incorrect. This function is decreasing rather than increasing.
Function is defined by , where and are constants. In the xy-plane, the graph of has a y-intercept at . The product of and is . What is the value of ?
The correct answer is . It’s given that . Substituting for in the equation yields . It’s given that the y-intercept of the graph of is . Substituting for and for in the equation yields , which is equivalent to , or . Adding to both sides of this equation yields . It’s given that the product of and is , or . Substituting for in this equation yields . Dividing both sides of this equation by yields .
The function is defined by the given equation. For what value of does reach its minimum?
Choice B is correct. It's given that , which can be rewritten as . Since the coefficient of the -term is positive, the graph of in the xy-plane opens upward and reaches its minimum value at its vertex. For an equation in the form , where , , and are constants, the x-coordinate of the vertex is . For the equation , , , and . It follows that the x-coordinate of the vertex is , or . Therefore, reaches its minimum when the value of is .
Choice A is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of in the xy-plane.
Choice C is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of in the xy-plane.
Choice D is incorrect. This is the y-coordinate of the vertex of the graph of in the xy-plane.
An entomologist recommended a program to reduce a certain invasive beetle population in an area. The given function estimates this beetle species' population years after , where . Which of the following is the best interpretation of in this context?
The estimated initial beetle population for this species and area in
The estimated beetle population for this species and area years after
The estimated percent decrease in the beetle population for this species and area each year after
The estimated percent decrease in the beetle population for this species and area every years after
Choice A is correct. For an exponential function in the form , where and are positive constants and , the initial value of , or the value of when , is and the value of decreases by each time increases by . Therefore, the initial value of the function , or the value of when , is . Therefore, the best interpretation of in this context is the estimated initial beetle population for this species and area in .
Choice B is incorrect. The estimated beetle population for this species and area years after is , or approximately , not .
Choice C is incorrect. The estimated percent decrease in the beetle population for this species and area each year after is , or , not .
Choice D is incorrect. The estimated percent decrease in the beetle population for this species and area every years after is , or approximately , not .
The function gives the number of bacteria in a population minutes after an initial observation. How much time, in minutes, does it take for the number of bacteria in the population to double?
Choice B is correct. It’s given that minutes after an initial observation, the number of bacteria in a population is . This expression consists of the initial number of bacteria, , multiplied by the expression . The time, in minutes, it takes for the number of bacteria to double is the increase in the value of that causes the expression to double. Since the base is , the expression will double when the exponent increases by . Since the exponent of this expression is , the exponent will increase by when increases by . Therefore, the time, in minutes, it takes for the number of bacteria in the population to double is .
Choice A is incorrect. This is the base of the exponent, not the time it takes for the number of bacteria in the population to double.
Choice C is incorrect. This is the number of minutes it takes for the population to double twice.
Choice D is incorrect. This is the number of bacteria that are initially observed, not the time it takes for the number of bacteria in the population to double.
Scientists recorded data about the ocean water levels at a certain location over a period of hours. The graph shown models the data, where represents sea level. Which table gives values of and their corresponding values of based on the model?
Choice C is correct. Each point on the graph represents an elapsed time , in hours, and the corresponding ocean water level , in feet, at a certain location based on the model. The graph shown passes through the points , , and . Thus, the table in choice C gives the values of and their corresponding values of based on the model.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
What is the positive solution to the given equation?
Choice A is correct. The given absolute value equation can be rewritten as two linear equations: and , or . Subtracting from both sides of the equation yields . Subtracting from both sides of the equation yields . Thus, the given equation has two possible solutions, and . Therefore, the positive solution to the given equation is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The function gives the predicted population, in millions, of a certain country for the period from to , where is the number of years after . According to the model, what is the best interpretation of the statement “ is approximately equal to ”?
In , the predicted population of this country was approximately million.
In , the predicted population of this country was approximately million.
years after , the predicted population of this country was approximately million.
years after , the predicted population of this country was approximately million.
Choice C is correct. The function gives the predicted population, in millions, of a certain country for the period from to , where is the number of years after . Since the value of is the value of when , it follows that " is approximately equal to " means that the value of is approximately equal to when . Therefore, the best interpretation of the statement " is approximately equal to " is that years after , the predicted population of this country was approximately million.
Choice A is incorrect. In , the predicted population of this country was million, not approximately million.
Choice B is incorrect. In , the predicted population of this country was million, not approximately million.
Choice D is incorrect. years after , the predicted population of this country was million, or approximately million, not approximately million.
The given equation relates the variables , , and . Which equation correctly expresses in terms of and ?
Choice D is correct. Subtracting from both sides of the given equation yields . Dividing both sides of this equation by yields . Therefore, the equation correctly expresses in terms of and .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
A company has a newsletter. In January , there were customers subscribed to the newsletter. For the next months after January , the total number of customers subscribed to the newsletter each month was greater than the total number subscribed the previous month. Which equation gives the total number of customers, , subscribed to the company's newsletter months after January , where ?
Choice B is correct. It's given that in January , there were customers subscribed to a company's newsletter and for the next months after January , the total number of customers subscribed to the newsletter each month was greater than the total number subscribed the previous month. It follows that this situation can be represented by the equation , where is the total number of customers subscribed to the company's newsletter months after January , is the number of customers subscribed to the newsletter in January , and the total number of customers subscribed to the newsletter each month was greater than the total number subscribed the previous month. Substituting for and for in this equation yields , or .
Choice A is incorrect. This equation represents a situation where the total number of customers subscribed each month was less, not greater, than the total number subscribed the previous month.
Choice C is incorrect. This equation represents a situation where the total number of customers subscribed each month was , not , greater than the total number subscribed the previous month.
Choice D is incorrect. This equation represents a situation where the total number of customers subscribed each month was , not , greater than the total number subscribed the previous month.
If x is a solution to the given equation, which of the following is a possible value of ?
Choice A is correct. The given equation can be rewritten as . Multiplying both sides of this equation by
yields
. Dividing both sides of this equation by 4 yields
. Taking the square root of both sides of this equation yields
or
. Therefore, a possible value of
is
.
Choices B, C, and D are incorrect and may result from computational or conceptual errors.
The function is defined by . What is the value of ?
Choice B is correct. The value of is the value of when . It's given that the function is defined by . Substituting for in this equation yields . Since the positive square root of is , it follows that this equation can be rewritten as , or . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the value of , not .
Choice D is incorrect. This is the value of , not .
In the given equation, is a constant. The equation has exactly one real solution. What is the minimum possible value of ?
The correct answer is . It's given that . Squaring both sides of this equation yields , which is equivalent to the given equation if . It follows that if a solution to the equation satisfies , then it's also a solution to the given equation; if not, it's extraneous. The equation can be rewritten as . Adding to both sides of this equation yields . Subtracting from both sides of this equation yields . The number of solutions to a quadratic equation in the form , where , , and are constants, can be determined by the value of the discriminant, . Substituting for , for , and for in yields , or . The equation has exactly one real solution if the discriminant is equal to zero, or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, if , then the equation has exactly one real solution. Substituting for in this equation yields , or , which is equivalent to . Taking the square root of both sides of this equation yields . Adding to both sides of this equation yields . To check whether this solution satisfies , the solution, , can be substituted for in , which yields , or . Since is greater than , it follows that if , or , then the given equation has exactly one real solution. If , then the discriminant, , is negative and the given equation has no solutions. Therefore, the minimum possible value of is .
In the xy-plane, when the graph of the function , where , is shifted up units, the resulting graph is defined by the function . If the graph of crosses through the point , where is a constant, what is the value of ?
The correct answer is . It's given that in the xy-plane, when the graph of the function , where , is shifted up units, the resulting graph is defined by the function . Therefore, function can be defined by the equation . It's given that . Substituting for in the equation yields . For the point , the value of is . Substituting for in the equation yields , or . It follows that the graph of crosses through the point . Therefore, the value of is .
The function is defined by . What is the value of ?
Choice C is correct. It’s given that the function is defined by . Substituting for in the given function yields , which is equivalent to , or . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the value of , not .
Which of the following expressions is equivalent to ?
Choice D is correct. Since is a common factor of each term in the given expression, the expression can be rewritten as .
Choice A is incorrect. This expression is equivalent to .
Choice B is incorrect. This expression is equivalent to .
Choice C is incorrect. This expression is equivalent to .
A rectangular volleyball court has an area of 162 square meters. If the length of the court is twice the width, what is the width of the court, in meters?
9
18
27
54
Choice A is correct. It’s given that the volleyball court is rectangular and has an area of 162 square meters. The formula for the area of a rectangle is , where A is the area,
is the length, and w is the width of the rectangle. It’s also given that the length of the volleyball court is twice the width, thus
. Substituting the given value into the formula for the area of a rectangle and using the relationship between length and width for this rectangle yields
. This equation can be rewritten as
. Dividing both sides of this equation by 2 yields
. Taking the square root of both sides of this equation yields
. Since the width of a rectangle is a positive number, the width of the volleyball court is 9 meters.
Choice B is incorrect because this is the length of the rectangle. Choice C is incorrect because this is the result of using 162 as the perimeter rather than the area. Choice D is incorrect because this is the result of calculating w in the equation instead of
.
A machine launches a softball from ground level. The softball reaches a maximum height of meters above the ground at seconds and hits the ground at seconds. Which equation represents the height above ground , in meters, of the softball seconds after it is launched?
Choice D is correct. An equation representing the height above ground , in meters, of a softball seconds after it is launched by a machine from ground level can be written in the form , where , , and are positive constants. In this equation, represents the time, in seconds, at which the softball reaches its maximum height of meters above the ground. It's given that this softball reaches a maximum height of meters above the ground at seconds; therefore, and . Substituting for and for in the equation yields . It's also given that this softball hits the ground at seconds; therefore, when . Substituting for and for in the equation yields , which is equivalent to , or . Adding to both sides of this equation yields . Dividing both sides of this equation by yields . Substituting for in the equation yields . Therefore, represents the height above ground , in meters, of this softball seconds after it is launched.
Choice A is incorrect. This equation represents a situation where the maximum height is meters above the ground at seconds, not meters above the ground at seconds.
Choice B is incorrect. This equation represents a situation where the maximum height is meters above the ground at seconds, not seconds.
Choice C is incorrect and may result from conceptual or calculation errors.
The function is defined by , where and are constants. In the -plane, the graph of has an -intercept at and a -intercept at . What is the value of ?
The correct answer is . It's given that the function is defined by , where and are constants. It's also given that the graph of has a y-intercept at . It follows that . Substituting for and for in yields , or . Subtracting from each side of this equation yields . Therefore, the value of is .
For the exponential function , the value of is , where is a constant. Of the following equations that define the function , which equation shows the value of as the coefficient or the base?
Choice B is correct. Each of the given choices is an equation of the form , where , , and are constants. For an equation of this form, the coefficient, , is equal to the value of the function when the exponent is equal to , or when . It follows that in the equation , the coefficient, , is equal to the value of . Substituting for in this equation yields , which is equivalent to , or . Thus, the value of is and the equation shows the value of as the coefficient.
Choice A is incorrect. This equation shows the value of , not , as the coefficient.
Choice C is incorrect. This equation shows the value of , not , as the coefficient.
Choice D is incorrect. This equation shows the value of , not , as the coefficient.
The function S above models the annual salary, in dollars, of an employee n years after starting a job, where a is a constant. If the employee’s salary increases by 4% each year, what is the value of a ?
0.04
0.4
1.04
1.4
Choice C is correct. A model for a quantity S that increases by a certain percentage per time period n is an exponential function in the form , where I is the initial value at time
for r% annual increase. It’s given that the annual increase in an employee’s salary is 4%, so
. The initial value can be found by substituting 0 for n in the given function, which yields
. Therefore,
. Substituting these values for r and I into the form of the exponential function
yields
, or
. Therefore, the value of a in the given function is 1.04.
Choices A, B, and D are incorrect and may result from incorrectly representing the annual increase in the exponential function.
During a 5-second time interval, the average acceleration a, in meters per second squared, of an object with an initial velocity of 12 meters per second is defined by the equation , where vf is the final velocity of the object in meters per second. If the equation is rewritten in the form vf = xa + y, where x and y are constants, what is the value of x ?
The correct answer is 5. The given equation can be rewritten in the form , like so:
It follows that the value of x is 5 and the value of y is 12.
The expression is equivalent to , where is a constant. What is the value of ?
The correct answer is . Applying the distributive property to the expression yields . Since is equivalent to , it follows that is also equivalent to . Since these expressions are equivalent, it follows that corresponding coefficients are equivalent. Therefore, and . Solving either of these equations for will yield the value of . Dividing both sides of by yields . Therefore, the value of is .
The revenue , in dollars, that a company receives from sales of a product is given by the function f above, where x is the unit price, in dollars, of the product. The graph of
in the xy-plane intersects the x-axis at 0 and a. What does a represent?
The revenue, in dollars, when the unit price of the product is $0
The unit price, in dollars, of the product that will result in maximum revenue
The unit price, in dollars, of the product that will result in a revenue of $0
The maximum revenue, in dollars, that the company can make
Choice C is correct. By definition, the y-value when a function intersects the x-axis is 0. It’s given that the graph of the function intersects the x-axis at 0 and a, that x is the unit price, in dollars, of a product, and that , where
, is the revenue, in dollars, that a company receives from the sales of the product. Since the value of a occurs when
, a is the unit price, in dollars, of the product that will result in a revenue of $0.
Choice A is incorrect. The revenue, in dollars, when the unit price of the product is $0 is represented by , when
. Choice B is incorrect. The unit price, in dollars, of the product that will result in maximum revenue is represented by the x-coordinate of the maximum of f. Choice D is incorrect. The maximum revenue, in dollars, that the company can make is represented by the y-coordinate of the maximum of f.
A conservation scientist implemented a program to reduce the population of a certain species in an area. The given function estimates this species' population years after , where . Which of the following is the best interpretation of in this context?
The estimated percent decrease in the population for this species and area every years after
The estimated percent decrease in the population for this species and area each year after
The estimated population for this species and area years after
The estimated initial population for this species and area in
Choice D is correct. Substituting for in the given equation yields , which is equivalent to , or . It’s given that the function estimates the species’ population years after , so it follows that the estimated population of the species is in . Therefore, the best interpretation of in this context is the estimated initial population for this species and area in .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect. The estimated percent decrease in the population for this species and area each year after is , not .
Choice C is incorrect. The estimated population for this species and area years after is , or approximately , not .
The equation above is true for all , where r and t are positive constants. What is the value of rt ?
15
20
60
Choice C is correct. To express the sum of the two rational expressions on the left-hand side of the equation as the single rational expression on the right-hand side of the equation, the expressions on the left-hand side must have the same denominator. Multiplying the first expression by results in
, and multiplying the second expression by
results in
, so the given equation can be rewritten as
, or
. Since the two rational expressions on the left-hand side of the equation have the same denominator as the rational expression on the right-hand side of the equation, it follows that
. Combining like terms on the left-hand side yields
, so it follows that
and
. Therefore, the value of
is
.
Choice A is incorrect and may result from an error when determining the sign of either r or t. Choice B is incorrect and may result from not distributing the 2 and 3 to their respective terms in . Choice D is incorrect and may result from a calculation error.
Which of the following is an equivalent form of ?
Choice C is correct. The first expression can be rewritten as
. Applying the distributive property to this product yields
. This difference can be rewritten as
. Distributing the factor of
through the second expression yields
. Regrouping like terms, the expression becomes
. Combining like terms yields
.
Choices A, B, and D are incorrect and likely result from errors made when applying the distributive property or combining the resulting like terms.
| Day | Number of bacteria per milliliter at end of day |
| 1 |
|
| 2 |
|
| 3 |
|
A culture of bacteria is growing at an exponential rate, as shown in the table above. At this rate, on which day would the number of bacteria per milliliter reach ?
Day 5
Day 9
Day 11
Day 12
Choice D is correct. The number of bacteria per milliliter is doubling each day. For example, from day 1 to day 2, the number of bacteria increased from 2.5 × 105 to 5.0 × 105. At the end of day 3 there are 106 bacteria per milliliter. At the end of day 4, there will be 106 × 2 bacteria per milliliter. At the end of day 5, there will be (106 × 2) × 2, or 106 × (22) bacteria per milliliter, and so on. At the end of day d, the number of bacteria will be 106 × (2d – 3). If the number of bacteria per milliliter will reach 5.12 × 108 at the end of day d, then the equation must hold. Since 5.12 × 108 can be rewritten as 512 × 106, the equation is equivalent to
. Rewriting 512 as 29 gives d – 3 = 9, so d = 12. The number of bacteria per milliliter would reach 5.12 × 108 at the end of day 12.
Choices A, B, and C are incorrect. Given the growth rate of the bacteria, the number of bacteria will not reach 5.12 × 108 per milliliter by the end of any of these days.
Which expression is equivalent to ?
Choice A is correct. The given expression can be rewritten as , which is equivalent to , or .
Choice B is incorrect. This expression is equivalent to .
Choice C is incorrect. This expression is equivalent to .
Choice D is incorrect and may result from conceptual or calculation errors.
One solution to the given equation can be written as , where is a constant. What is the value of ?
The correct answer is . The solutions to a quadratic equation of the form can be calculated using the quadratic formula and are given by . The given equation is in the form , where , , and . It follows that the solutions to the given equation are , which is equivalent to , or . It's given that one solution to the equation can be written as . The solution is in this form. Therefore, the value of is .
The equation above estimates the global data traffic D, in terabytes, for the year that is t years after 2010. What is the best interpretation of the number 5,640 in this context?
The estimated amount of increase of data traffic, in terabytes, each year
The estimated percent increase in the data traffic, in terabytes, each year
The estimated data traffic, in terabytes, for the year that is t years after 2010
The estimated data traffic, in terabytes, in 2010
Choice D is correct. Since t represents the number of years after 2010, the estimated data traffic, in terabytes, in 2010 can be calculated using the given equation when . Substituting 0 for t in the given equation yields
, or
. Thus, 5,640 represents the estimated data traffic, in terabytes, in 2010.
Choice A is incorrect. Since the equation is exponential, the amount of increase of data traffic each year isn’t constant. Choice B is incorrect. According to the equation, the percent increase in data traffic each year is 90%. Choice C is incorrect. The estimated data traffic, in terabytes, for the year that is t years after 2010 is represented by D, not the number 5,640.
The graphs of the given equations in the xy-plane intersect at the point . What is a possible value of ?
Choice B is correct. Since the point is an intersection point of the graphs of the given equations in the xy-plane, the pair should satisfy both equations, and thus is a solution of the given system. According to the first equation, . Substituting in place of in the second equation yields . Adding to both sides of this equation yields . Taking the square root of both sides of this equation yields two solutions: and . Of these two solutions, only is given as a choice.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the value of coordinate , rather than , of the intersection point .
The given equation relates the positive numbers , , and . Which equation correctly gives in terms of and ?
Choice B is correct. It's given that the equation relates the positive numbers , , and . Dividing both sides of the given equation by yields . Subtracting from both sides of this equation yields , or . It follows that the equation correctly gives in terms of and .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
For what value of is the given expression equivalent to , where ?
The correct answer is . An expression of the form , where is an integer greater than and , is equivalent to . Therefore, the given expression, where , is equivalent to . Applying properties of exponents, this expression can be rewritten as , or , which can be rewritten as , or . It's given that the expression is equivalent to , where . It follows that is equivalent to . Therefore, . Dividing both sides of this equation by yields , or . Thus, the value of for which the given expression is equivalent to , where , is . Note that 4/225, .0177, .0178, 0.017, and 0.018 are examples of ways to enter a correct answer.
Which of the following is a solution to the equation above?
2
Choice D is correct. A quadratic equation in the form , where a, b, and c are constants, can be solved using the quadratic formula:
. Subtracting
from both sides of the given equation yields
. Applying the quadratic formula, where
,
, and
, yields
. This can be rewritten as
. Since
, or
, the equation can be rewritten as
. Dividing 2 from both the numerator and denominator yields
or
. Of these two solutions, only
is present among the choices. Thus, the correct choice is D.
Choice A is incorrect and may result from a computational or conceptual error. Choice B is incorrect and may result from using instead of
as the quadratic formula. Choice C is incorrect and may result from rewriting
as
instead of
.
In the given quadratic function, and are constants. The graph of in the xy-plane is a parabola that opens upward and has a vertex at the point , where and are constants. If and , which of the following must be true?
I only
II only
I and II
Neither I nor II
Choice D is correct. It's given that the graph of in the xy-plane is a parabola with vertex . If , then for the graph of , the point with an x-coordinate of and the point with an x-coordinate of have the same y-coordinate. In the xy-plane, a parabola is a symmetric graph such that when two points have the same y-coordinate, these points are equidistant from the vertex, and the x-coordinate of the vertex is halfway between the x-coordinates of these two points. Therefore, for the graph of , the points with x-coordinates and are equidistant from the vertex, , and is halfway between and . The value that is halfway between and is , or . Therefore, . The equation defining can also be written in vertex form, . Substituting for in this equation yields , or . This equation is equivalent to , or . Since , it follows that and . Dividing both sides of the equation by yields , or . Since , it's not true that . Therefore, statement II isn't true. Substituting for in the equation yields , or . Subtracting from both sides of this equation yields . If , then , or . Since could be any value less than , it's not necessarily true that . Therefore, statement I isn't necessarily true. Thus, neither I nor II must be true.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The x-intercept of the graph shown is . What is the value of ?
The correct answer is . It’s given that the x-intercept of the graph shown is . The graph passes through the point . Therefore, the value of is .
How many distinct real solutions does the given equation have?
Exactly one
Exactly two
Infinitely many
Zero
Choice D is correct. Since the square of a real number is never negative, the given equation isn't true for any real value of . Therefore, the given equation has zero distinct real solutions.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
In the xy-plane, a line with equation for some constant intersects a parabola at exactly one point. If the parabola has equation , what is the value of ?
The correct answer is . The given linear equation is . Dividing both sides of this equation by yields . Substituting for in the equation of the parabola yields . Adding and to both sides of this equation yields . Since it’s given that the line and the parabola intersect at exactly one point, the equation must have exactly one solution. An equation of the form , where , , and are constants, has exactly one solution when the discriminant, , is equal to . In the equation , where , , and , the discriminant is . Setting the discriminant equal to yields , or . Adding to both sides of this equation yields . Dividing both sides of this equation by yields . Note that 81/4 and 20.25 are examples of ways to enter a correct answer.
Which of the following could be the equation of the graph shown in the xy-plane?
Choice B is correct. Each of the given choices is an equation of the form , where , , , and are positive constants. In the xy-plane, the graph of an equation of this form has x-intercepts at , , and . The graph shown has x-intercepts at , , and . Therefore, and . Of the given choices, only choices A and B have and . For an equation in the form , if all values of that are less than or greater than correspond to negative y-values, then the sum of all the exponents of the factors on the right-hand side of the equation is even. In the graph shown, all values of less than or greater than correspond to negative y-values. Therefore, the sum of all the exponents of the factors on the right-hand side of the equation must be even. For choice A, the sum of these exponents is , or , which is odd. For choice B, the sum of these exponents is , or , which is even. Therefore, could be the equation of the graph shown.
Choice A is incorrect. For the graph of this equation, all values of less than correspond to positive, not negative, y-values.
Choice C is incorrect. The graph of this equation has x-intercepts at , , and , rather than x-intercepts at , , and .
Choice D is incorrect. The graph of this equation has x-intercepts at , , and , rather than x-intercepts at , , and .
The function models the population, in thousands, of a certain city years after . According to the model, the population is predicted to increase by every months. What is the value of ?
Choice A is correct. It’s given that the function models the population, in thousands, of a certain city years after . The value of the base of the given exponential function, , corresponds to an increase of for every increase of in the exponent, . If the exponent is equal to , then . Multiplying both sides of this equation by yields . If the exponent is equal to , then . Multiplying both sides of this equation by yields , or . Therefore, the population is predicted to increase by every of a year. It’s given that the population is predicted to increase by every months. Since there are months in a year, of a year is equivalent to , or , months. Therefore, the value of is .
Choice B is incorrect. This is the number of months in which the population is predicted to increase by according to the model , not .
Choice C is incorrect. This is the number of months in which the population is predicted to increase by according to the model , not .
Choice D is incorrect. This is the number of months in which the population is predicted to increase by according to the model , not .
Which ordered pair is a solution to the given system of equations?
Choice B is correct. The second equation in the given system is . Substituting for in the first equation of the given system yields . Subtracting from both sides of this equation yields . Subtracting from both sides of this equation yields . Factoring out a common factor of from the left-hand side of this equation yields . It follows that or . Dividing both sides of the equation by yields . Substituting for in the equation yields , or . Therefore, a solution to the given system of equations is the ordered pair .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
What is a solution to the given equation?
The correct answer is either or . Adding to each side of the given equation yields . Since a product of two factors is equal to if and only if at least one of the factors is , either or . Adding to each side of the equation yields . Subtracting from each side of the equation yields . Therefore, the solutions to the given equation are and . Note that -30 and 30 are examples of ways to enter a correct answer.
The function is defined by . For what value of does reach its minimum?
Choice A is correct. It's given that . Since , it follows that . Expanding the quantity in this equation yields . Distributing the and the yields . Combining like terms yields . Therefore, . For a quadratic function defined by an equation of the form , where , , and are constants and is positive, reaches its minimum, , when the value of is . The equation can be rewritten in this form by completing the square. This equation is equivalent to , or . This equation can be rewritten as , or , which is equivalent to . This equation is in the form , where , , and . Therefore, reaches its minimum when the value of is .
Choice B is incorrect. This is the value of for which , rather than , reaches its minimum.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the value of for which , rather than , reaches its minimum.
The graph of is shown, where the function is defined by and , , , and are constants. For how many values of does ?
One
Two
Three
Four
Choice C is correct. If a value of satisfies , the graph of will contain a point and thus touch the x-axis. Since there are points at which this graph touches the x-axis, there are values of for which .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Which expression is equivalent to ?
Choice B is correct. The given expression can be rewritten as . Since the two terms of this expression are both constant multiples of , they are like terms and can, therefore, be combined through addition. Adding like terms in the expression yields .
Choice A is incorrect. This is equivalent to , not .
Choice C is incorrect. This is equivalent to , not .
Choice D is incorrect. This is equivalent to , not .
Which expression is equivalent to ?
Choice D is correct. The given expression shows addition of two like terms. Therefore, the given expression is equivalent to , or .
Choice A is incorrect. This expression is equivalent to , not .
Choice B is incorrect. This expression is equivalent to , not .
Choice C is incorrect. This expression is equivalent to , not .
| x | y |
| 0 | 0 |
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
The table shown includes some values of x and their corresponding values of y. Which of the following graphs in the xy-plane could represent the relationship between x and y ?
Choice B is correct. Each pair of values shown in the table gives the ordered pair of coordinates for a point that lies on the graph that represents the relationship between x and y in the xy-plane: ,
,
, and
. Only the graph in choice B passes through the points listed in the table that are visible in the given choices.
Choices A, C, and D are incorrect. None of these graphs passes through the point .
The expression is equivalent to the expression , where , , and are constants. What is the value of ?
The correct answer is . It's given that the expression is equivalent to the expression , where , , and are constants. Applying the distributive property to the given expression, , yields , which can be rewritten as . Combining like terms yields . Since this expression is equivalent to , it follows that the value of is .
The graph of is shown, where and are constants. What is the value of ?
The correct answer is . Since the graph passes through the point , it follows that when the value of is , the value of is . Substituting for and for in the given equation yields , or . Therefore, the value of is . Substituting for in the given equation yields . Since the graph passes through the point , it follows that when the value of is , the value of is . Substituting for and for in the equation yields , or , which is equivalent to . Adding to each side of this equation yields . Dividing each side of this equation by yields . Since the value of is and the value of is , it follows that the value of is , or .
Alternate approach: The given equation represents a parabola in the xy-plane with a vertex at . Therefore, the given equation, , which is written in standard form, can be written in vertex form, , where is the vertex of the parabola and is the value of the coefficient on the term when the equation is written in standard form. It follows that . Substituting for , for , and for in this equation yields , or . Squaring the binomial on the right-hand side of this equation yields . Multiplying each term inside the parentheses on the right-hand side of this equation by yields , which is equivalent to . From the given equation , it follows that the value of is and the value of is . Therefore, the value of is , or .
What is the sum of the solutions to the given equation?
The correct answer is . Taking the square root of each side of the given equation yields or . Adding to both sides of the equation yields . Adding to both sides of the equation yields . Therefore, the sum of the solutions to the given equation is , or .
The graph models the number of active projects a company was working on months after the end of November , where . According to the model, what is the predicted number of active projects the company was working on at the end of November ?
Choice B is correct. It's given that the graph models the number of active projects a company was working on months after the end of November . Therefore, the value of that corresponds to the end of November is . The point at which is the y-intercept of the graph. It follows that the y-intercept of the graph shown is the point . Therefore, according to the model, the predicted number of active projects the company was working on at the end of November is .
Choice A is incorrect. This is the value of that corresponds to the end of November , not the predicted number of active projects the company was working on at the end of November .
Choice C is incorrect. This is the predicted number of active projects the company was working on months after the end of November .
Choice D is incorrect. This is the predicted number of active projects the company was working on months after the end of November .
Which expression is equivalent to ?
Choice B is correct. Combining like terms inside the parentheses of the given expression, , yields . Combining like terms in this resulting expression yields .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Which expression is equivalent to ?
Choice D is correct. The expression can be rewritten as . To add the two terms of this expression, the terms can be rewritten with a common denominator. Since , the expression can be rewritten as . Since , the expression can be rewritten as . Therefore, the expression can be rewritten as , which is equivalent to . Applying the distributive property to each term of the numerator yields , or . Adding like terms in the numerator yields .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The function is defined by . In the xy-plane, the graph of is the result of translating the graph of up units. What is the value of ?
The correct answer is . It's given that the graph of is the result of translating the graph of up units in the xy-plane. It follows that the graph of is the same as the graph of . Substituting for in the equation yields . It's given that . Substituting for in the equation yields . Substituting for in this equation yields , or . Thus, the value of is .
Which of the following is equivalent to ?
Choice C is correct. Using the distributive property to multiply 3 and gives
, which can be rewritten as
.
Choice A is incorrect and may result from rewriting the given expression as . Choice B is incorrect and may result from incorrectly rewriting the expression as
. Choice D is incorrect and may result from incorrectly rewriting the expression as
.
The graph of a system of a linear equation and a nonlinear equation is shown. What is the solution to this system?
Choice C is correct. The solution to the system of two equations corresponds to the point where the graphs of the equations intersect. The graphs of the linear equation and the nonlinear equation shown intersect at the point . Thus, the solution to the system is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
What is the product of the solutions to the given equation?
8
3
Choice D is correct. By the zero-product property, if , then
,
, or
. Solving each of these equations for
yields
,
, or
. The product of these solutions is
.
Choice A is incorrect and may result from sign errors made when solving the given equation. Choice B is incorrect and may result from finding the sum, not the product, of the solutions. Choice C is incorrect and may result from finding the sum, not the product, of the solutions in addition to making sign errors when solving the given equation.
In the expression above, b and c are positive integers. If the expression is equivalent to and
, which of the following could be the value of c ?
4
6
8
10
Choice A is correct. If the given expression is equivalent to , then
, where x isn’t equal to b. Multiplying both sides of this equation by
yields
. Since the right-hand side of this equation is in factored form for the difference of squares, the value of c must be a perfect square. Only choice A gives a perfect square for the value of c.
Choices B, C, and D are incorrect. None of these values of c produces a difference of squares. For example, when 6 is substituted for c in the given expression, the result is . The expression
can’t be factored with integer values, and therefore
isn’t equivalent to
.
What is the solution to the equation above?
0
2
3
5
Choice B is correct. Since is equivalent to 1, the right-hand side of the given equation can be rewritten as
, or
. Since the left- and right-hand sides of the equation
have the same denominator, it follows that
. Applying the distributive property of multiplication to the expression
yields
, or
. Therefore,
. Subtracting x and 2 from both sides of this equation yields
.
Choices A, C, and D are incorrect. If , then
. This can be rewritten as
, which is a false statement. Therefore, 0 isn’t a solution to the given equation. Substituting 3 and 5 into the given equation yields similarly false statements.
The functions and are defined by the equations shown. Which expression is equivalent to ?
Choice D is correct. It’s given that and . Substituting for and for in the expression yields . This expression can be rewritten as , or , which is equivalent to .
Choice A is incorrect. This expression is equivalent to , not .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This expression is equivalent to , not .
A rectangle has a length of units and a width of units. If the rectangle has an area of square units, what is the value of ?
Choice B is correct. The area of a rectangle is equal to its length multiplied by its width. Multiplying the given length, units, by the given width, units, yields square units. If the rectangle has an area of square units, it follows that , or . Subtracting from both sides of this equation yields . Factoring the left-hand side of this equation yields . Applying the zero product property to this equation yields two solutions: and . Since is the rectangle’s length, in units, which must be positive, the value of is .
Choice A is incorrect. This is the width, in units, of the rectangle, not the value of .
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the area, in square units, of the rectangle, not the value of .
A scientist initially measures bacteria in a growth medium. hours later, the scientist measures bacteria. Assuming exponential growth, the formula gives the number of bacteria in the growth medium, where and are constants and is the number of bacteria hours after the initial measurement. What is the value of ?
Choice B is correct. It’s given that the formula gives the number of bacteria in a growth medium, where and are constants and is the number of bacteria hours after the initial measurement. It’s also given that a scientist initially measures bacteria in the growth medium. Since the initial measurement is hours after the initial measurement, it follows that when , . Substituting for and for in the given equation yields , or , which is equivalent to . It’s given that hours later, the scientist measures bacteria, or when , . Substituting for , for , and for in the given equation yields . Dividing each side of this equation by yields , or , which is equivalent to . Dividing both sides of this equation by yields . Therefore, the value of is .
Choice A is incorrect. This is the value of the reciprocal of .
Choice C is incorrect. This is the value of the reciprocal of .
Choice D is incorrect. This is the value of .
What is the smallest solution to the given equation?
The correct answer is . Squaring both sides of the given equation yields , which can be rewritten as . Subtracting and from both sides of this equation yields . This quadratic equation can be rewritten as . According to the zero product property, equals zero when either or . Solving each of these equations for yields or . Therefore, the given equation has two solutions, and . Of these two solutions, is the smallest solution to the given equation.
If the ordered pair satisfies the system of equations above, what is one possible value of x ?
The correct answer is either 0 or 3. For an ordered pair to satisfy a system of equations, both the x- and y-values of the ordered pair must satisfy each equation in the system. Both expressions on the right-hand side of the given equations are equal to y, therefore it follows that both expressions on the right-hand side of the equations are equal to each other: . This equation can be rewritten as
, and then through factoring, the equation becomes
. Because the product of the two factors is equal to 0, it can be concluded that either
or
, or rather,
or
. Note that 0 and 3 are examples of ways to enter a correct answer.
The solution to the given system of equations is . What is the value of ?
Choice B is correct. Adding to each side of the second equation in the given system of equations yields . Substituting for in the first equation yields . Subtracting from each side of this equation yields . This equation can be rewritten as . Taking the square root of each side of this equation yields . Subtracting from each side of this equation yields . Therefore, the value of is .
Choice A is incorrect. This is the value of , not .
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
A quadratic function models a projectile's height, in meters, above the ground in terms of the time, in seconds, after it was launched. The model estimates that the projectile was launched from an initial height of meters above the ground and reached a maximum height of meters above the ground seconds after the launch. How many seconds after the launch does the model estimate that the projectile will return to a height of meters?
Choice B is correct. It's given that a quadratic function models the projectile's height, in meters, above the ground in terms of the time, in seconds, after it was launched. It follows that an equation representing the model can be written in the form , where is the projectile's estimated height above the ground, in meters, seconds after the launch, is a constant, and is the maximum height above the ground, in meters, the model estimates the projectile reached seconds after the launch. It's given that the model estimates the projectile reached a maximum height of meters above the ground seconds after the launch. Therefore, and . Substituting for and for in the equation yields . It's also given that the model estimates that the projectile was launched from an initial height of meters above the ground. Therefore, when , . Substituting for and for in the equation yields , or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Substituting for in the equation yields . Therefore, the equation models the projectile's height, in meters, above the ground seconds after it was launched. The number of seconds after the launch that the model estimates that the projectile will return to a height of meters is the value of when . Substituting for in yields . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Taking the square root of both sides of this equation yields two equations: and . Adding to both sides of the equation yields . Adding to both sides of the equation yields . Since seconds after the launch represents the time at which the projectile was launched, must be the number of seconds the model estimates that the projectile will return to a height of meters.
Alternate approach: It's given that a quadratic function models the projectile's height, in meters, above the ground in terms of the time, in seconds, after it was launched. It's also given that the model estimates that the projectile was launched from an initial height of meters above the ground and reached a maximum height of meters above the ground seconds after the launch. Since the model is quadratic, and quadratic functions are symmetric, the model estimates that for any given height less than the maximum height, the time the projectile takes to travel from the given height to the maximum height is the same as the time the projectile takes to travel from the maximum height back to the given height. Thus, since the model estimates the projectile took seconds to travel from meters above the ground to its maximum height of meters above the ground, the model also estimates the projectile will take more seconds to travel from its maximum height of meters above the ground back to meters above the ground. Thus, the model estimates that the projectile will return to a height of meters seconds after it reaches its maximum height, which is seconds after the launch.
Choice A is incorrect. The model estimates that seconds after the launch, the projectile reached a height of meters, not meters.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The given equation relates the variables and . For what value of does the value of reach its minimum?
The correct answer is . When an equation is of the form , where , , and are constants, the value of reaches its minimum when . Since the given equation is of the form , it follows that , , and . Therefore, the value of reaches its minimum when , or .
Which of the following expressions is equivalent to the expression above?
Choice B is correct. One of the properties of radicals is . Thus, the given expression can be rewritten as
. Simplifying by taking the cube root of each part gives x1 ⋅ y2, or xy2.
Choices A, C, and D are incorrect and may be the result of incorrect application of the properties of exponents and radicals.
Which expression is equivalent to ?
Choice C is correct. Applying the distributive property to the given expression yields . Applying the distributive property once again to this expression yields , or . This expression can be rewritten as . Thus, is equivalent to .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
An oceanographer uses the equation to model the speed s, in knots, of an ocean wave, where p represents the period of the wave, in seconds. Which of the following represents the period of the wave in terms of the speed of the wave?
Choice A is correct. It’s given that p represents the period of the ocean wave, so the equation can be solved for p to represent the period of the wave in terms of the speed of the wave. Multiplying both sides of the equation by the reciprocal of
will isolate p. This yields
, which simplifies to
. Therefore,
.
Choices B, C, and D are incorrect and may result from errors made when rearranging the equation to solve for p.
If , what is the value of ?
Choice B is correct. Dividing each side of the given equation by yields . Squaring both sides of this equation yields . Multiplying each side of this equation by yields . Therefore, the value of is .
Choice A is incorrect. This is the value of , not .
Choice C is incorrect. This is the value of , not .
Choice D is incorrect. This is the value of , not .
The function is defined by . What is the value of when ?
Choice D is correct. It's given that . Substituting for in this equation yields , or . Therefore, when , the value of is .
Choice A is incorrect. This is the value of when .
Choice B is incorrect. This is the value of when .
Choice C is incorrect. This is the value of when .
If and
, which of the following is equal to
?
Choice B is correct. It’s given that and
. Using the distributive property, the expression
can be rewritten as
. Substituting
and
for
and
, respectively, in this expression yields
. Expanding this expression yields
. Combining like terms, this expression can be rewritten as
.
Choices A, C, and D are incorrect and may result from an error in using the distributive property, substituting, or combining like terms.
For the given function , the graph of in the xy-plane passes through the point , where is a constant. What is the value of ?
The correct answer is . It's given that the graph of in the xy-plane passes through the point , where is a constant. It follows that equals . Substituting for in the given function yields , or . Therefore, the value of is .
Which of the following is a solution to the equation ?
1
2
3
4
Choice B is correct. Subtracting x2 from both sides of the given equation yields x2 – 4 = 0. Adding 4 to both sides of the equation gives x2 = 4. Taking the square root of both sides of the equation gives x = 2 or x = –2. Therefore, x = 2 is one solution to the original equation.
Alternative approach: Subtracting x2 from both sides of the given equation yields x2 – 4 = 0. Factoring this equation gives x2 – 4 = (x + 2)(x – 2) = 0, such that x = 2 or x = –2. Therefore, x = 2 is one solution to the original equation.
Choices A, C, and D are incorrect and may be the result of computation errors.
The given equation relates the positive numbers , , and . Which equation correctly expresses in terms of and ?
Choice B is correct. Adding to each side of the given equation yields . Therefore, the equation correctly expresses in terms of and .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
A park ranger hung squirrel houses each in the shape of a right rectangular prism for fox squirrels. Each house has a height of inches. The length of each house's base is inches, which is inch more than the width of the house's base. Which function gives the volume of each house, in cubic inches, in terms of the length of the house's base?
Choice A is correct. The volume of a prism is equal to the area of its base times its height. It's given that the length of each house's base is inches and that this length is inch more than the width, in inches, of the house's base. It follows that the width, in inches, of the house's base is . The area of a rectangle is the product of its length and its width. Therefore, the area of the base of the house is square inches. It's given that the height of each house is inches. Therefore, the function that gives the volume of each house, in cubic inches, in terms of the length of the house's base is , or .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
In the given equation, and are positive constants. The product of the solutions to the given equation is , where is a constant. What is the value of ?
Choice A is correct. The left-hand side of the given equation is the expression . Applying the distributive property to this expression yields . Since the first two terms of this expression have a common factor of and the last two terms of this expression have a common factor of , this expression can be rewritten as . Since the two terms of this expression have a common factor of , it can be rewritten as . Therefore, the given equation can be rewritten as . By the zero product property, it follows that or . Subtracting from both sides of the equation yields . Subtracting from both sides of the equation yields . Dividing both sides of this equation by yields . Therefore, the solutions to the given equation are and . It follows that the product of the solutions of the given equation is , or . It’s given that the product of the solutions of the given equation is . It follows that , which can also be written as . It’s given that and are positive constants. Therefore, dividing both sides of the equation by yields . Thus, the value of is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
In the given equation, is a positive integer. The equation has no real solution. What is the greatest possible value of ?
The correct answer is . A quadratic equation of the form , where , , and are constants, has no real solution if and only if its discriminant, , is negative. In the given equation, and . Substituting for and for in this expression yields a discriminant of , or . Since this value must be negative, , or . Taking the positive square root of each side of this inequality yields . Since is a positive integer, and the greatest integer less than is , the greatest possible value of is .
Function is defined by , where and are constants. In the xy-plane, the graph of has a y-intercept at . The product of and is . What is the value of ?
The correct answer is . It’s given that . Substituting for in the equation yields . It’s given that the y-intercept of the graph of is . Substituting for and for in the equation yields , which is equivalent to , or . Adding to both sides of this equation yields . It’s given that the product of and is , or . Substituting for in this equation yields . Dividing both sides of this equation by yields .
The given equation relates the distinct positive numbers , , and . Which equation correctly expresses in terms of and ?
Choice A is correct. To express in terms of and , the given equation must be solved for . Since it's given that is a positive number, is not equal to zero. Therefore, multiplying both sides of the given equation by yields the equivalent equation . Since it's given that is a positive number, is not equal to zero. Therefore, dividing each side of the equation by yields the equivalent equation .
Choice B is incorrect. This equation is equivalent to .
Choice C is incorrect. This equation is equivalent to .
Choice D is incorrect. This equation is equivalent to .
If , which of the following expressions is equivalent to
?
Choice D is correct. Taking the square root of an exponential expression halves the exponent, so , which further reduces to
. This can be rewritten as
.
Choice A is incorrect and may result from neglecting the denominator of the given expression and from incorrectly calculating the square root of 16. Choice B is incorrect and may result from rewriting as
rather than
. Choice C is incorrect and may result from taking the square root of the variables in the numerator twice instead of once.
The function is defined by , where is a constant. In the xy-plane, the graph of passes through the point . What is the value of ?
The correct answer is . By the zero product property, if , then , which gives , or , which gives . Therefore, when and when . Since the graph of passes through the point , it follows that , so . Substituting for in the equation yields . The value of can be calculated by substituting for in this equation, which yields , or .
What is the minimum value of the given function?
Choice B is correct. For a quadratic function defined by an equation of the form , where , , and are constants and , the minimum value of the function is . In the given function, , , and . Therefore, the minimum value of the given function is .
Choice A is incorrect. This is the value of for which the given function reaches its minimum value, not the minimum value of the function.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The area of a rectangular banner is square inches. The banner's length , in inches, is inches longer than its width, in inches. Which equation represents this situation?
Choice A is correct. It’s given that the banner’s length , in inches, is inches longer than its width, in inches. It follows that the banner’s width, in inches, can be represented by the expression . The area of a rectangle is the product of its length and its width. It's given that the area of the banner is square inches, so it follows that , or . Subtracting from each side of this equation yields . Therefore, the equation that represents this situation is .
Choice B is incorrect and may result from representing the width, in inches, of the banner as , rather than .
Choice C is incorrect and may result from representing the width, in inches, of the banner as , rather than .
Choice D is incorrect and may result from conceptual or calculation errors.
The function is defined by the given equation. For what value of does reach its minimum?
Choice D is correct. It's given that , which can be rewritten as . Since the coefficient of the -term is positive, the graph of in the xy-plane opens upward and reaches its minimum value at its vertex. The x-coordinate of the vertex is the value of such that reaches its minimum. For an equation in the form , where , , and are constants, the x-coordinate of the vertex is . For the equation , , , and . It follows that the x-coordinate of the vertex is , or . Therefore, reaches its minimum when the value of is .
Alternate approach: The value of for the vertex of a parabola is the x-value of the midpoint between the two x-intercepts of the parabola. Since it’s given that , it follows that the two x-intercepts of the graph of in the xy-plane occur when and , or at the points and . The midpoint between two points, and , is . Therefore, the midpoint between and is , or . It follows that reaches its minimum when the value of is .
Choice A is incorrect. This is the y-coordinate of the y-intercept of the graph of in the xy-plane.
Choice B is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of in the xy-plane.
Choice C is incorrect and may result from conceptual or calculation errors.
If , what is the value of ?
The correct answer is . Dividing each side of the given equation by yields . Adding to each side of this equation yields . Therefore, if , the value of is .
In the xy-plane, what is the y-coordinate of the point of intersection of the graphs of and
?
The correct answer is 1. The point of intersection of the graphs of the given equations is the solution to the system of the two equations. Since and
, it follows that
, or
. Applying the distributive property to the left-hand side of this equation yields
. Subtracting
from and adding 3 to both sides of this equation yields
. Factoring the left-hand side of this equation yields
. By the zero product property, if
, it follows that
. Adding 2 to both sides of
yields
. Substituting 2 for x in either of the given equations yields
. For example, substituting 2 for x in the second given equation yields
, or
. Therefore, the point of intersection of the graphs of the given equations is
. The y-coordinate of this point is 1.
A sample of a certain isotope takes years to decay to half its original mass. The function gives the approximate mass of this isotope, in grams, that remains years after a -gram sample starts to decay. Which statement is the best interpretation of in this context?
Approximately grams of the sample remains years after the sample starts to decay.
The mass of the sample has decreased by approximately grams years after the sample starts to decay.
The mass of the sample has decreased by approximately grams years after the sample starts to decay.
Approximately grams of the sample remains years after the sample starts to decay.
Choice A is correct. In the given function, represents the approximate mass, in grams, of the sample that remains years after the sample starts to decay. It follows that the best interpretation of is that approximately grams of the sample remains years after the sample starts to decay.
Choice B is incorrect. The mass of the sample has decreased by approximately , or , grams, not grams, years after the sample starts to decay.
Choice C is incorrect. The mass of the sample has decreased by approximately grams, not grams, years after the sample starts to decay.
Choice D is incorrect. This would be the best interpretation of , not .
In the given system of equations, is a constant. The system has exactly one distinct real solution. What is the value of ?
The correct answer is . Subtracting the second equation from the first equation yields , or . This is equivalent to . It's given that the system has exactly one distinct real solution; therefore, this equation has exactly one distinct real solution. An equation of the form , where , , and are constants, has exactly one distinct real solution when the discriminant, , is equal to . The equation is of this form, where , , and . Substituting these values into the discriminant, , yields . Setting the discriminant equal to yields , or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Note that 35/2 and 17.5 are examples of ways to enter a correct answer.
The functions and are defined by the given equations.
If , where is a constant, what is the value of ?
The correct answer is . The value of is the value of when . Substituting for in the equation yields , or , which is equivalent to , or . Since it's given that , it follows that and the value of is the value of . Substituting for in the equation yields , or , which is equivalent to , or . Note that -4.9 and -49/10 are examples of ways to enter a correct answer.
Which quadratic equation has no real solutions?
Choice D is correct. The number of solutions to a quadratic equation in the form , where , , and are constants, can be determined by the value of the discriminant, . If the value of the discriminant is greater than zero, then the quadratic equation has two distinct real solutions. If the value of the discriminant is equal to zero, then the quadratic equation has exactly one real solution. If the value of the discriminant is less than zero, then the quadratic equation has no real solutions. For the quadratic equation in choice D, , , , and . Substituting for , for , and for in yields , or . Since is less than zero, it follows that the quadratic equation has no real solutions.
Choice A is incorrect. The value of the discriminant for this quadratic equation is . Since is greater than zero, it follows that this quadratic equation has two real solutions.
Choice B is incorrect. The value of the discriminant for this quadratic equation is . Since zero is equal to zero, it follows that this quadratic equation has exactly one real solution.
Choice C is incorrect. The value of the discriminant for this quadratic equation is . Since is greater than zero, it follows that this quadratic equation has two real solutions.
What is the y-intercept of the graph shown?
Choice B is correct. The y-intercept of a graph in the xy-plane is the point at which the graph crosses the y-axis. The graph shown crosses the y-axis at the point . Therefore, the y-intercept of the graph shown is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
What is the positive solution to the given equation?
The correct answer is . Dividing both sides of the given equation by yields . By the definition of absolute value, if , then or . Therefore, the two solutions to the equation are and . It follows that the positive solution to the given equation is .
In the equation above, t is a constant. If the equation has no real solutions, which of the following could be the value of t ?
1
3
Choice A is correct. The number of solutions to any quadratic equation in the form , where a, b, and c are constants, can be found by evaluating the expression
, which is called the discriminant. If the value of
is a positive number, then there will be exactly two real solutions to the equation. If the value of
is zero, then there will be exactly one real solution to the equation. Finally, if the value of
is negative, then there will be no real solutions to the equation.
The given equation is a quadratic equation in one variable, where t is a constant. Subtracting t from both sides of the equation gives
. In this form,
,
, and
. The values of t for which the equation has no real solutions are the same values of t for which the discriminant of this equation is a negative value. The discriminant is equal to
; therefore,
. Simplifying the left side of the inequality gives
. Subtracting 16 from both sides of the inequality and then dividing both sides by 8 gives
. Of the values given in the options,
is the only value that is less than
. Therefore, choice A must be the correct answer.
Choices B, C, and D are incorrect and may result from a misconception about how to use the discriminant to determine the number of solutions of a quadratic equation in one variable.
The function gives the estimated number of marine mammals in a certain area, where is the number of years since a study began. What is the best interpretation of in this context?
The estimated number of marine mammals in the area was when the study began.
The estimated number of marine mammals in the area was when the study began.
The estimated number of marine mammals in the area increased by each year during the study.
The estimated number of marine mammals in the area increased by each year during the study.
Choice B is correct. The function gives the estimated number of marine mammals in a certain area, where is the number of years since a study began. Since the value of is the value of when , it follows that means that the value of is when . Since is the number of years since the study began, it follows that is years since the study began, or when the study began. Therefore, the best interpretation of in this context is the estimated number of marine mammals in the area was when the study began.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
What is the x-intercept of the graph shown?
Choice D is correct. The x-intercept of the graph shown is the point on the graph where . At , the corresponding value of is . Therefore, the x-intercept of the graph shown is .
Choice A is incorrect. This is the x-intercept of a graph in the xy-plane that intersects the x-axis at , not .
Choice B is incorrect. This is the x-intercept of a graph in the xy-plane that intersects the x-axis at , not .
Choice C is incorrect. This is the x-intercept of a graph in the xy-plane that intersects the x-axis at , not .
Which table gives three values of and their corresponding values of for function ?
Choice D is correct. Substituting for in the given function yields , which is equivalent to , or . Therefore, when , the corresponding value of for function is . Substituting for in the given function yields , which is equivalent to , or . Therefore, when , the corresponding value of for function is . Substituting for in the given function yields , which is equivalent to , or . Therefore, when , the corresponding value of for function is . Of the choices given, only the table in choice D gives these three values of and their corresponding values of for function .
Choice A is incorrect. This table gives three values of and their corresponding values of for the function .
Choice B is incorrect. This table gives three values of and their corresponding values of for the function .
Choice C is incorrect and may result from conceptual or calculation errors.
The given equation relates the numbers , , and . Which equation correctly expresses in terms of and ?
Choice A is correct. Subtracting from each side of the given equation results in . Dividing each side of this equation by results in .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
If the given function is graphed in the xy-plane, where , what is the x-coordinate of an x-intercept of the graph?
The correct answer is either or . The x-intercepts of a graph in the xy-plane are the points where . Thus, for an x-intercept of the graph of , . Substituting for in the equation yields . By the zero product property, and . Subtracting from both sides of the equation yields . Adding to both sides of the equation yields . Therefore, the x-coordinates of the x-intercepts of the graph of are and . Note that -6 and 4 are examples of ways to enter a correct answer.
The solutions to are and , where . The solutions to are and , where . The solutions to , where is a constant, are and . What is the value of ?
The correct answer is . Subtracting from both sides of the equation yields . To complete the square, adding , or , to both sides of this equation yields , or . Taking the square root of both sides of this equation yields . Subtracting from both sides of this equation yields . Therefore, the solutions and to the equation are and . Since , it follows that and . Subtracting from both sides of the equation yields . To complete the square, adding , or , to both sides of this equation yields , or . Taking the square root of both sides of this equation yields , or . Subtracting from both sides of this equation yields . Therefore, the solutions and to the equation are and . Since , it follows that and . It's given that the solutions to , where is a constant, are and . It follows that this equation can be written as , which is equivalent to . Therefore, the value of is . Substituting for , for , for , and for in this equation yields , which is equivalent to , or , which is equivalent to , or . Therefore, the value of is .
Which expression is equivalent to ?
Choice C is correct. Since is a common factor of each term in the given expression, the given expression can be rewritten as .
Choice A is incorrect. This expression is equivalent to .
Choice B is incorrect. This expression is equivalent to .
Choice D is incorrect. This expression is equivalent to .
In the given equation, is an integer constant. If the equation has no real solution, what is the least possible value of ?
The correct answer is . An equation of the form , where , , and are constants, has no real solutions if and only if its discriminant, , is negative. Applying the distributive property to the left-hand side of the equation yields . Adding to each side of this equation yields . Substituting for , for , and for in yields a discriminant of , or . If the given equation has no real solution, it follows that the value of must be negative. Therefore, . Adding to both sides of this inequality yields . Dividing both sides of this inequality by yields , or . Since it's given that is an integer, the least possible value of is .
What is the sum of the solutions to the given equation?
The correct answer is . Subtracting from each side of the given equation yields . Since is a common factor of each of the terms on the right-hand side of this equation, it can be rewritten as . This is equivalent to , or . Dividing both sides of this equation by yields . Since a product of two factors is equal to if and only if at least one of the factors is , either or . Subtracting from both sides of the equation yields . Adding to both sides of the equation yields . Dividing both sides of this equation by yields . Therefore, the solutions to the given equation are and . It follows that the sum of the solutions to the given equation is , which is equivalent to , or . Note that 31/3 and 10.33 are examples of ways to enter a correct answer.
At the time of posting a video, a social media channel had subscribers. Each day for five days after the video was posted, the number of subscribers doubled from the number the previous day. Which equation gives the total number of subscribers, , to the channel days after the video was posted?
Choice B is correct. It's given that each day for five days after a social media channel posted a video, the number of subscribers doubled from the number the previous day. Since the number of subscribers doubled each day, the relationship between and can be represented by an exponential equation of the form , where is the number of subscribers at the time of posting the video and the number of subscribers to the channel increases by a factor of each day. It's given that at the time of posting the video, the channel had subscribers. Therefore, . It's also given that the number of subscribers doubled, or increased by a factor of , from the number the previous day. Therefore, . Substituting for and for in the equation yields .
Choice A is incorrect. This equation gives the total number of subscribers to a channel for which the initial number of subscribers was and the number increased each day by times the number from the previous day.
Choice C is incorrect. This equation gives the total number of subscribers to a channel for which the number of subscribers each day was half the number from the previous day, rather than double the number.
Choice D is incorrect and may result from conceptual errors.
The function is defined by . What is the value of ?
Choice C is correct. It’s given that function is defined by the equation . The value of is the value of when . Substituting for in the given equation yields , which is equivalent to , or .
Choice A is incorrect. This is the value of when , rather than .
Choice B is incorrect. This is the value of when , rather than .
Choice D is incorrect. This is the value of when , rather than .
What is a positive solution to the given equation?
Choice C is correct. Applying the zero product property to the given equation yields three equations: , , and . Subtracting from both sides of the equation yields . Adding to both sides of the equation yields . Subtracting from both sides of the equation yields . Therefore, the solutions to the given equation are , , and . It follows that a positive solution to the given equation is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
In the given equation, is a constant. The equation has exactly one solution. What is the value of ?
Choice B is correct. It's given that the equation , where is a constant, has exactly one solution. A quadratic equation of the form has exactly one solution if and only if its discriminant, , is equal to zero. It follows that for the given equation, and . Substituting for and for in yields , or . Since the discriminant must equal zero, it follows that . Subtracting from both sides of this equation yields . Dividing each side of this equation by yields . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
What is the positive solution to the given equation?
Choice B is correct. Multiplying each side of the given equation by yields . To complete the square, adding to each side of this equation yields , or . Taking the square root of each side of this equation yields two equations: and . Subtracting from each side of the equation yields . Dividing each side of this equation by yields , or . Therefore, is a solution to the given equation. Subtracting from each side of the equation yields . Dividing each side of this equation by yields . Therefore, the given equation has two solutions, and . Since is positive, it follows that is the positive solution to the given equation.
Alternate approach: Adding and to each side of the given equation yields . The right-hand side of this equation can be rewritten as . Factoring out the common factor of from the first two terms of this expression and the common factor of from the second two terms yields . Factoring out the common factor of from these two terms yields the expression . Since this expression is equal to , it follows that either or . Adding to each side of the equation yields . Dividing each side of this equation by yields . Therefore, is a positive solution to the given equation. Subtracting from each side of the equation yields . Therefore, the given equation has two solutions, and . Since is positive, it follows that is the positive solution to the given equation.
Choice A is incorrect. Substituting for in the given equation yields , which is false.
Choice C is incorrect. Substituting for in the given equation yields , which is false.
Choice D is incorrect. Substituting for in the given equation yields , which is false.
Which expression is equivalent to ?
Choice A is correct. The expression can be rewritten as , which is equivalent to .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Immanuel purchased a certain rare coin on January 1. The function , where , gives the predicted value, in dollars, of the rare coin years after Immanuel purchased it. What is the best interpretation of the statement “ is approximately equal to ” in this context?
When the rare coin's predicted value is approximately dollars, it is greater than the predicted value, in dollars, on January 1 of the previous year.
When the rare coin’s predicted value is approximately dollars, it is times the predicted value, in dollars, on January 1 of the previous year.
From the day Immanuel purchased the rare coin to years after Immanuel purchased the coin, its predicted value increased by a total of approximately dollars.
years after Immanuel purchased the rare coin, its predicted value is approximately dollars.
Choice D is correct. It’s given that the function gives the predicted value, in dollars, of a certain rare coin years after Immanuel purchased it. It follows that represents the predicted value, in dollars, of the coin years after Immanuel purchased it. Since the value of is the value of when , it follows that “ is approximately equal to ” means that is approximately equal to when . Therefore, the best interpretation of the statement “ is approximately equal to ” in this context is years after Immanuel purchased the rare coin, its predicted value is approximately dollars.
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
The function gives the product of a number, , and a number that is more than . Which equation defines ?
Choice C is correct. It’s given that the function gives the product of a number, , and a number that is more than . A number that is more than can be represented by the expression . Therefore, can be defined by the equation , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
A model estimates that at the end of each year from to , the number of squirrels in a population was more than the number of squirrels in the population at the end of the previous year. The model estimates that at the end of , there were squirrels in the population. Which of the following equations represents this model, where is the estimated number of squirrels in the population years after the end of and ?
Choice B is correct. Since the model estimates that the number of squirrels in the population increased by a fixed percentage, , each year, the model can be represented by an exponential equation of the form , where is the estimated number of squirrels in the population at the end of , and the model estimates that at the end of each year, the number is more than the number at the end of the previous year. Since the model estimates that at the end of each year, the number was more than the number at the end of the previous year, . Substituting for in the equation yields , which is equivalent to , or . It’s given that the estimated number of squirrels at the end of was . This means that when , . Substituting for and for in the equation yields , or . Dividing each side of this equation by yields . Substituting for in the equation yields .
Choice A is incorrect. This equation represents a model where at the end of each year, the estimated number of squirrels was of, not more than, the estimated number at the end of the previous year.
Choice C is incorrect. This equation represents a model where at the end of each year, the estimated number of squirrels was of, not more than, the estimated number at the end of the previous year, and the estimated number of squirrels at the end of , not the end of , was .
Choice D is incorrect. This equation represents a model where the estimated number of squirrels at the end of , not the end of , was .
The given expression can be rewritten as , where is a constant. What is the value of ?
The correct answer is . It's given that the expression can be rewritten as . Applying the distributive property to the expression yields . Therefore, can be rewritten as . It follows that in the expressions and , the coefficients of are equivalent, the coefficients of are equivalent, and the constant terms are equivalent. Therefore, , , and . Solving any of these equations for yields the value of . Dividing both sides of the equation by yields . Therefore, the value of is . Note that .09 and 9/100 are examples of ways to enter a correct answer.
The function is defined by the given equation. For which of the following values of does ?
Choice A is correct. The value of for which can be found by substituting for and for in the given equation, , which yields . For this equation to be true, either or . Adding to both sides of the equation yields . Dividing both sides of this equation by yields . To check whether is the value of , substituting for in the equation yields , which is equivalent to , or , which isn't a true statement. Therefore, isn't the value of . Adding to both sides of the equation yields . Dividing both sides of this equation by yields . To check whether is the value of , substituting for in the equation yields , which is equivalent to , or , which is a true statement. Therefore, the value of for which is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The given equation relates the distinct positive real numbers , , and . Which equation correctly expresses in terms of and ?
Choice C is correct. Dividing each side of the given equation by yields , or . Because it's given that each of the variables is positive, squaring each side of this equation yields the equivalent equation . Subtracting from each side of this equation yields , or .
Choice A is incorrect. This equation isn't equivalent to the given equation.
Choice B is incorrect. This equation isn't equivalent to the given equation.
Choice D is incorrect. This equation isn't equivalent to the given equation.
Which of the following is equivalent to the expression above?
(x + 3)2 + 5
(x + 3)2 – 5
(x – 3)2 + 5
(x – 3)2 – 5
Choice B is correct. The given quadratic expression is in standard form, and each answer choice is in vertex form. Completing the square converts the expression from standard form to vertex form. The first step is to rewrite the expression as follows: . The first three terms of the revised expression can be rewritten as a perfect square as follows:
. Combining the constant terms gives
.
Choice A is incorrect. Squaring the binomial and simplifying the expression in choice A gives . Combining like terms gives
, not
. Choice C is incorrect. Squaring the binomial and simplifying the expression in choice C gives
. Combining like terms gives
, not
. Choice D is incorrect. Squaring the binomial and simplifying the expression in choice D gives
. Combining like terms gives
, not
.
In the expression , p is a constant. This expression is equivalent to the expression
. What is the value of p ?
Choice B is correct. Using the distributive property, the first given expression can be rewritten as 6x2 + 3px + 24 – 16px – 64x + 24, and then rewritten as 6x2 + (3p – 16p – 64)x + 24. Since the expression 6x2 + (3p – 16p – 64)x + 24 is equivalent to 6x2 – 155x + 24, the coefficients of the x terms from each expression are equivalent to each other; thus 3p – 16p – 64 = –155. Combining like terms gives –13p – 64 = –155. Adding 64 to both sides of the equation gives –13p = –71. Dividing both sides of the equation by –13 yields p = 7.
Choice A is incorrect. If p = –3, then the first expression would be equivalent to 6x2 – 25x + 24. Choice C is incorrect. If p = 13, then the first expression would be equivalent to 6x2 – 233x + 24. Choice D is incorrect. If p = 155, then the first expression would be equivalent to 6x2 – 2,079x + 24.
The function is defined by the given equation. The function is defined by . Which expression represents the maximum value of ?
Choice B is correct. It’s given that function is defined by and that . Substituting for in yields , or . The maximum value of can be found by completing the square to rewrite the equation defining in the form , where the maximum value of the function is , which occurs when , and is a negative constant. The equation is equivalent to , which can be rewritten as , or . This equation is in the form , where , , and . Thus, the maximum value of is .
Alternate approach: Since the function is a quadratic function, the maximum value of occurs at the value of that’s halfway between the two zeros of the function. The zeros of function can be found by substituting for in the equation defining , which yields . This equation can be rewritten as . By the zero product property, it follows that or . Subtracting from each side of the equation yields . Dividing each side of this equation by yields . Therefore, the zeros of function are and . The value that’s halfway between and can be found by calculating the average of and , which is , or . It follows that the maximum of function occurs when . Substituting for in the equation defining function yields , which is equivalent to . Multiplying by in this equation to get a common denominator yields , or , which is equivalent to . Thus, the maximum value of is . Since the equation defining is , the maximum value of is greater than the maximum value of . It follows that the maximum value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Which of the following is a solution to the given equation?
1
0
Choice A is correct. Applying the distributive property on the left- and right-hand sides of the given equation yields , or
. Subtracting
from and adding
to both sides of this equation yields
. Subtracting 6 from both sides of this equation and then dividing both sides by 10 yields
.
Choices B, C, and D are incorrect. Substituting 0, , or
for x in the given equation will result in a false statement. If
, the given equation becomes
; if
, the given equation becomes
; and if
, the given equation becomes
. Therefore, the values 0,
, and
aren’t solutions to the given equation.
Which graph represents the given system of equations?
Choice A is correct. The graph of a quadratic equation in the form has its vertex at . The first equation in the given system of equations is , so the graph of this quadratic equation has its vertex at . The graph of a linear equation of the form has a slope of and a y-intercept at . The second equation in the given system of equations is , so the graph of this linear equation has a slope of and a y-intercept at . Of the choices, only choice A has the graph of a quadratic equation with its vertex at and the graph of a linear equation with a slope of and a y-intercept at .
Choice B is incorrect. This graph represents a system in which the second equation is , not .
Choice C is incorrect. This graph represents a system in which the first equation is , not .
Choice D is incorrect. This graph represents a system in which the first equation is , not , and the second equation is , not .
In the equation above, a is a constant and . If the equation has two integer solutions, what is a possible value of a ?
The correct answer is either 7, 8, or 13. Since the given equation has two integer solutions, the expression on the left-hand side of this equation can be factored as , where c and d are also integers. The product of c and d must equal the constant term of the original quadratic expression, which is 12. Additionally, the sum of c and d must be a negative number since it’s given that
, but the sign preceding a in the given equation is negative. The possible pairs of values for c and d that satisfy both of these conditions are
and
,
and
, and
and
. Since the value of
is the sum of c and d, the possible values of
are
,
, and
. It follows that the possible values of a are 7, 8, and 13. Note that 7, 8, and 13 are examples of ways to enter a correct answer.
The function estimates an object’s height, in feet, above the ground seconds after the object is dropped, where is a constant. The function estimates that the object is feet above the ground when it is dropped at . Approximately how many seconds after being dropped does the function estimate the object will hit the ground?
Choice B is correct. It's given that the function estimates that the object is feet above the ground when it's dropped at . Substituting for and for in the function yields , or . Substituting for in the function yields . When the object hits the ground, its height will be feet above the ground. Substituting for in yields . Adding to each side of this equation yields . Dividing each side of this equation by yields . Since the object will hit the ground at a positive number of seconds after it's dropped, the value of can be found by taking the positive square root of each side of this equation, which yields . It follows that the function estimates the object will hit the ground approximately seconds after being dropped.
Choice A is incorrect. The function estimates that seconds after being dropped, the object's height will be feet, or feet, above the ground.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
From 2005 through 2014, the number of music CDs sold in the United States declined each year by approximately 15% of the number sold the preceding year. In 2005, approximately 600 million CDs were sold in the United States. Of the following, which best models C, the number of millions of CDs sold in the United States, t years after 2005?
Choice B is correct. A model for a quantity C that decreases by a certain percentage per time period t is an exponential equation in the form , where I is the initial value at time
for r% annual decline. It’s given that C is the number of millions of CDs sold in the United States and that t is the number of years after 2005. It’s also given that 600 million CDs were sold at time
, so
. This number declines by 15% per year, so
. Substituting these values into the equation produces
, or
.
Choice A is incorrect and may result from errors made when representing the percent decline. Choices C and D are incorrect. These equations model exponential increases in CD sales, not exponential decreases.
What is the solution set of the equation above?
Choice B is correct. Subtracting 4 from both sides of isolates the radical expression on the left side of the equation as follows:
. Squaring both sides of
yields
. This equation can be rewritten as a quadratic equation in standard form:
. One way to solve this quadratic equation is to factor the expression
by identifying two numbers with a sum of
and a product of
. These numbers are
and 1. So the quadratic equation can be factored as
. It follows that 5 and
are the solutions to the quadratic equation. However, the solutions must be verified by checking whether 5 and
satisfy the original equation,
. When
, the original equation gives
, or
, which is false. Therefore,
does not satisfy the original equation. When
, the original equation gives
, or
, which is true. Therefore,
is the only solution to the original equation, and so the solution set is
.
Choices A, C, and D are incorrect because each of these sets contains at least one value that results in a false statement when substituted into the given equation. For instance, in choice D, when 0 is substituted for x into the given equation, the result is , or
. This is not a true statement, so 0 is not a solution to the given equation.
The graph of the function f, defined by , is shown in the xy-plane above. If the function g (not shown) is defined by
, what is one possible value of a such that
?
The correct answer is either 2 or 8. Substituting in the definitions for f and g gives
and
, respectively. If
, then
. Subtracting 10 from both sides of this equation gives
. Multiplying both sides by
gives
. Expanding
gives
. Combining the like terms on one side of the equation gives
. One way to solve this equation is to factor
by identifying two numbers with a sum of
and a product of 16. These numbers are
and
, so the quadratic equation can be factored as
. Therefore, the possible values of a are either 2 or 8. Note that 2 and 8 are examples of ways to enter a correct answer.
Alternate approach: Graphically, the condition implies the graphs of the functions
and
intersect at
. The graph
is given, and the graph of
may be sketched as a line with y-intercept 10 and a slope of
(taking care to note the different scales on each axis). These two graphs intersect at
and
.
The expression is equivalent to , where b is a constant and . What is the value of b?
Choice A is correct. Since the given expressions are equivalent and the numerator of the second expression is of the numerator of the first expression, the denominator of the second expression must also be of the denominator of the first expression. By the distributive property, is equivalent to , or . Therefore, the value of is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
In the given equation, is a constant. The equation has no real solutions if . What is the least possible value of ?
The correct answer is . A quadratic equation of the form , where , , and are constants, has no real solutions when the value of the discriminant, , is less than . In the given equation, , and . Therefore, the discriminant of the given equation can be expressed as , or . It follows that the given equation has no real solutions when . Adding to both sides of this inequality yields . Dividing both sides of this inequality by yields , or . It's given that the equation has no real solutions when . Therefore, the least possible value of is .
Which of the following expressions has a factor of , where is a positive integer constant?
Choice D is correct. Since each choice has a term of , which can be written as , and each choice has a term of , which can be written as , the expression that has a factor of , where is a positive integer constant, can be represented as . Using the distributive property of multiplication, this expression is equivalent to , or . Combining the x-terms in this expression yields . It follows that the coefficient of the x-term is equal to . Thus, from the given choices, must be equal to , , , or . Therefore, must be equal to , , , or , respectively, and must be equal to , , , or , respectively. Of these four values of , only , or , is a positive integer. It follows that must be equal to because this is the only choice for which the value of is a positive integer constant. Therefore, the expression that has a factor of is .
Choice A is incorrect. If this expression has a factor of , then the value of is , which isn't positive.
Choice B is incorrect. If this expression has a factor of , then the value of is , which isn't an integer.
Choice C is incorrect. If this expression has a factor of , then the value of is , which isn't an integer.
An object was launched upward from a platform. The graph shown models the height above ground, , in meters, of the object seconds after it was launched. For which of the following intervals of time was the height of the object increasing for the entire interval?
From to
From to
From to
From to
Choice A is correct. It's given that the variable represents the height, in meters, of the object above the ground. The graph shows that the height of the object was increasing from to , and decreasing from to . Therefore, the height of the object was increasing for the entire interval of time from to .
Choice B is incorrect. The height of the object wasn't increasing for this entire interval of time, as it was decreasing from to .
Choice C is incorrect. The height of the object was decreasing, not increasing, for this entire interval of time.
Choice D is incorrect. The height of the object was decreasing, not increasing, for this entire interval of time.
The population P of a certain city y years after the last census is modeled by the equation below, where r is a constant and is the population when
.
If during this time the population of the city decreases by a fixed percent each year, which of the following must be true?
Choice B is correct. The term (1 + r) represents a percent change. Since the population is decreasing, the percent change must be between 0% and 100%. When the percent change is expressed as a decimal rather than as a percent, the percentage change must be between 0 and 1. Because (1 + r) represents percent change, this can be expressed as 0 < 1 + r < 1. Subtracting 1 from all three terms of this compound inequality results in –1 < r < 0.
Choice A is incorrect. If r < –1, then after 1 year, the population P would be a negative value, which is not possible. Choices C and D are incorrect. For any value of r > 0, 1 + r > 1, and the exponential function models growth for positive values of the exponent. This contradicts the given information that the population is decreasing.
Which expression is equivalent to ?
Choice A is correct. The given expression can be rewritten as
. Combining like terms yields
.
Choices B, C, and D are incorrect and may be the result of errors when applying the distributive property.
The graph of is shown in the xy-plane. What is the value of ?
Choice D is correct. Because the graph of is shown, the value of is the value of on the graph that corresponds with . When , the corresponding value of is . Therefore, the value of is .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
The function is defined by . What is the value of ?
Choice A is correct. The value of is the value of when . The function is defined by . Substituting for in this equation yields . This equation can be rewritten as , or . Therefore, the value of is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The graph gives the estimated number of catalogs , in thousands, a company sent to its customers at the end of each year, where represents the number of years since the end of , where . Which statement is the best interpretation of the y-intercept in this context?
The estimated total number of catalogs the company sent to its customers during the first years was .
The estimated total number of catalogs the company sent to its customers from the end of to the end of was .
The estimated number of catalogs the company sent to its customers at the end of was .
The estimated number of catalogs the company sent to its customers at the end of was .
Choice D is correct. The y-intercept of the graph is the point at which the graph crosses the y-axis, or the point for which the value of is . Therefore, the y-intercept of the given graph is the point . It's given that represents the number of years since the end of . Therefore, represents years since the end of , which is the same as the end of . It's also given that represents the estimated number of catalogs, in thousands, that the company sent to its customers at the end of the year. Therefore, represents catalogs. It follows that the y-intercept means that the estimated number of catalogs the company sent to its customers at the end of was .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The graph of is shown, where , and , , and are constants. For how many values of does ?
Three
Two
One
Zero
Choice D is correct. Each point on the graph of in the xy-plane gives a value of and its corresponding value of , or . For any value of for which , there is a corresponding point on the graph of with a y-coordinate of . A point with a y-coordinate of is a point where the graph intersects the x-axis. It's given that , where , , and are constants. In the xy-plane, the graph of an equation of this form will lie entirely either above or below the horizontal line defined by . The part of the graph of shown lies entirely below the horizontal line defined by , and thus the entire graph of must lie below the line defined by . It follows that the graph of will never intersect the x-axis. Therefore, there are zero values of x for which .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
The function is given. Which table of values represents ?
Choice B is correct. It’s given that and . Substituting for in the equation yields . Substituting for in this equation yields , or . Substituting for in the equation yields , or . Substituting for in the equation yields , or . Therefore, when then , when then , and when then . Thus, the table of values in choice B represents .
Choice A is incorrect. This table represents rather than .
Choice C is incorrect. This table represents rather than .
Choice D is incorrect. This table represents rather than .
What is one possible solution to the given equation?
The correct answer is or . By the definition of absolute value, if , then or . Adding to both sides of the first equation yields . Adding to both sides of the second equation yields . Thus, the given equation has two possible solutions, and . Note that 15 and -5 are examples of ways to enter a correct answer.
Which expression is equivalent to , where ?
Choice D is correct. Two fractions can be added together when they have a common denominator. Since , multiplying the second term in the given expression by yields , which is equivalent to . Therefore, the expression can be written as which is equivalent to . Since each term in the numerator of this expression has a factor of , the expression can be rewritten as , or , which is equivalent to .
Choice A is incorrect. This expression is equivalent to .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This expression is equivalent to .
A solution to the given system of equations is . What is the value of ?
Choice A is correct. The first equation in the given system of equations is . Substituting for in the second equation in the given system of equations yields , or . Substituting for and for in the expression yields , or . Therefore, the value of is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The function is defined by . What is the value of ?
The correct answer is . It’s given that the function is defined by . Substituting for in function yields , which is equivalent to , or . Therefore, the value of is .
The product of two positive integers is . If the first integer is greater than twice the second integer, what is the smaller of the two integers?
Choice B is correct. Let be the first integer and let be the second integer. If the first integer is greater than twice the second integer, then . If the product of the two integers is , then . Substituting for in this equation results in . Distributing the to both terms in the parentheses results in . Subtracting from both sides of this equation results in . The left-hand side of this equation can be factored by finding two values whose product is , or , and whose sum is . The two values whose product is and whose sum is are and . Thus, the equation can be rewritten as , which is equivalent to , or . By the zero product property, it follows that and . Subtracting from both sides of the equation yields . Dividing both sides of this equation by yields . Since is a positive integer, the value of is not . Adding to both sides of the equation yields . Substituting for in the equation yields . Dividing both sides of this equation by results in . Therefore, the two integers are and , so the smaller of the two integers is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the larger of the two integers.
Choice D is incorrect and may result from conceptual or calculation errors.
The function gives the value, in dollars, of a certain piece of equipment after months of use. If the value of the equipment decreases each year by of its value the preceding year, what is the value of ?
Choice C is correct. For a function of the form , where , , and are constants and , the value of decreases by for every increase of by . In the given function, , , and . Therefore, for the given function, the value of decreases by , or , for every increase of by . Since represents the value, in dollars, of the equipment after months of use, it follows that the value of the equipment decreases every months by of its value the preceding months. Since there are months in a year, the value of the equipment decreases each year by of its value the preceding year. Thus, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The given equation relates the variables , , and . Which equation correctly expresses in terms of and ?
Choice B is correct. Subtracting from each side of the given equation yields . Thus, the equation correctly expresses in terms of and .
Choice A is incorrect. This equation can be rewritten as .
Choice C is incorrect. This equation can be rewritten as .
Choice D is incorrect. This equation can be rewritten as .
In the given equation, is a constant. The equation has exactly one solution. What is the value of ?
The correct answer is . A quadratic equation in the form , where , , and are constants, has exactly one solution when its discriminant, , is equal to . In the given equation, , and . Substituting for and for in yields , or . Since the given equation has exactly one solution, . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, the value of is .
The expression , where
and
, is equivalent to which of the following?
Choice D is correct. For and
,
and
are equivalent to
and
, respectively. Also,
and
are equivalent to
and
, respectively. Therefore, the given expression can be rewritten as
.
Choices A, B, and C are incorrect because these choices are not equivalent to the given expression for and
.
For example, for and
, the value of the given expression is
; the values of the choices, however, are
,
, and 1, respectively.
In the xy-plane, the y-coordinate of the y-intercept of the graph of the function f is c. Which of the following must be equal to c ?
Choice A is correct. A y-intercept is the point in the xy-plane where the graph of the function crosses the y-axis, which is where . It’s given that the y-coordinate of the y-intercept of the graph of function f is c. It follows that the coordinate pair representing the y-intercept must be
. Therefore, c must equal
.
Choices B, C, and D are incorrect because ,
, and
would represent the y-value of the coordinate where
,
, and
, respectively.
The function models the intensity of an X-ray beam, in number of particles in the X-ray beam, millimeters below the surface of a sample of iron. According to the model, what is the estimated number of particles in the X-ray beam when it is at the surface of the sample of iron?
Choice A is correct. It's given that the function models the intensity of an X-ray beam, in number of particles in the X-ray beam, millimeters below the surface of a sample of iron. When the X-ray beam is at the surface of the sample of iron, it is millimeters below the surface, so the value of is . Substituting for in the function yields . Since any positive number raised to the power of is equal to , it follows that , or . Therefore, the estimated number of particles in the X-ray beam at the surface of the sample of iron is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
What is the positive solution to the given equation?
Choice C is correct. The left-hand side of the given equation can be factored as . Therefore, the given equation, , can be written as . Applying the zero product property to this equation yields and . Subtracting from both sides of the equation yields . Dividing both sides of this equation by yields . Adding to both sides of the equation yields . Therefore, the two solutions to the given equation, , are and . It follows that is the positive solution to the given equation.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The expression , where is a constant, can be rewritten as , where , , and are integer constants. Which of the following must be an integer?
Choice D is correct. It's given that can be rewritten as . The expression can be rewritten as , or . Therefore, is equivalent to . It follows that . Dividing each side of this equation by yields . Since is an integer, must be an integer. Therefore, must also be an integer.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The equation gives the estimated stock price , in dollars, for a certain company days after a new product launched, where . Which statement is the best interpretation of in this context?
The company's estimated stock price increased every day after the new product launched.
The company's estimated stock price increased every days after the new product launched.
day after the new product launched, the company's estimated stock price is .
days after the new product launched, the company's estimated stock price is .
Choice C is correct. In the given equation, represents the number of days after a new product launched, where , and represents the estimated stock price, in dollars, for a certain company. Therefore, the best interpretation of in this context is that day after the new product launched, the company's estimated stock price is .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
On April , there were views of an advertisement posted on a website. Every days after April , the number of views of the advertisement had increased by of the number of views days earlier. The function gives the predicted number of views days after April . Which equation defines ?
Choice C is correct. It’s given that on April , there were views of the advertisement. It’s also given that every days after April , the number of views of the advertisement had increased by of the number of views days earlier. This situation can be represented by an exponential function of the form , where is the number of views on April and every days after April , the number of views had increased by of the number of views days earlier. It follows that , , and . Substituting for , for , and for in the equation yields , or .
Choice A is incorrect. This function gives the predicted number of views for an advertisement for which every days, the number of views was , rather than increased by , of the number of views days earlier.
Choice B is incorrect. This function gives the predicted number of views for an advertisement for which every days, the number of views was of the number of views days earlier, rather than an advertisement for which every days, the number of views had increased by of the number of views days earlier.
Choice D is incorrect. This function gives the predicted number of views for an advertisement for which every days, rather than every days, the number of views had increased by of the number of views days earlier, rather than days earlier.
The graph shown models the number of residents of a certain city years after . How many residents does this model estimate the city had in ?
Choice C is correct. It's given that represents years after . Therefore, is represented by . On the model shown, the point with an x-coordinate of has a y-coordinate of . Thus, the model estimates that in , the city had residents.
Choice A is incorrect. This is the value of that represents the year .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is approximately the number of residents the model estimates the city had in , not .
How many distinct real solutions does the given equation have?
Exactly one
Exactly two
Infinitely many
Zero
Choice D is correct. Any quantity that is positive or negative in value has a positive value when squared. Therefore, the left-hand side of the given equation is either positive or zero for any value of . Since the right-hand side of the given equation is negative, there is no value of for which the given equation is true. Thus, the number of distinct real solutions for the given equation is zero.
Choices A, B, and C are incorrect and may result from conceptual errors.
What is the positive solution to the given equation?
The correct answer is . The given equation, , is equivalent to . Taking the square root of each side of this equation yields . Thus, the positive solution to the given equation is .
Which expression is a factor of ?
Choice A is correct. Since is a common factor of each of the terms in the given expression, the expression can be rewritten as . Therefore, the factors of the given expression are and . Of these two factors, only is listed as a choice.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is a term of the given expression, not a factor of the given expression.
Choice D is incorrect. This is a term of the given expression, not a factor of the given expression.
Which of the following expressions is equivalent to , for
?
Choice D is correct. Since and
, the fraction
can be written as
. It is given that
, so the common factor
is not equal to 0. Therefore, the fraction can be further simplified to
.
Choice A is incorrect. The expression is not equivalent to
because at
,
as a value of 1 and
has a value of 0.
Choice B is incorrect and results from omitting the factor x in the factorization of . Choice C is incorrect and may result from incorrectly factoring
as
instead of
.
What is one of the solutions to the given equation?
The correct answer is either or . The left-hand side of the given equation can be rewritten by factoring. The two values that multiply to and add to are and . It follows that the given equation can be rewritten as . Setting each factor equal to yields two equations: and . Subtracting from both sides of the equation results in . Adding to both sides of the equation results in . Note that 2 and -12 are examples of ways to enter a correct answer.
What is the x-coordinate of the x-intercept of the graph shown?
The correct answer is . An x-intercept of a graph is a point on the graph where it intersects the x-axis, or where the value of is . The graph shown intersects the x-axis at the point . Therefore, the x-coordinate of the x-intercept of the graph shown is .
If the given equations are graphed in the xy-plane, at how many points do the graphs of the equations intersect?
Exactly one
Exactly two
Infinitely many
Zero
Choice D is correct. A point is a solution to a system of equations if it lies on the graphs of both equations in the xy-plane. In other words, a solution to a system of equations is a point at which the graphs intersect. It’s given that the first equation is . Substituting for in the second equation yields . Subtracting from each side of this equation yields . Dividing each side of this equation by yields . Since the square of a real number is at least , this equation can't have any real solutions. Therefore, the graphs of the equations intersect at zero points.
Alternate approach: The graph of the second equation is a parabola that opens downward and has a vertex at . Therefore, the maximum value of this parabola occurs when . The graph of the first equation is a horizontal line at on the y-axis, or . Since is greater than , or the horizontal line is above the vertex of the parabola, the graphs of these equations intersect at zero points.
Choice A is incorrect. The graph of , not , and the graph of the second equation intersect at exactly one point.
Choice B is incorrect. The graph of any horizontal line such that the value of is less than , not greater than , and the graph of the second equation intersect at exactly two points.
Choice C is incorrect and may result from conceptual or calculation errors.
For the function f defined above, what is the value of ?
7
11
Choice C is correct. Substituting for x in the given function f gives
, which simplifies to
. This further simplifies to
, or
.
Choice A is incorrect and may result from correctly substituting for x in the function but incorrectly simplifying the resulting expression to
, or
. Choice B is incorrect and may result from arithmetic errors. Choice D is incorrect and may result from correctly substituting
for x in the function but incorrectly simplifying the expression to
, or 11.
Which of the following is the graph in the xy-plane of the given equation?
Choice D is correct. The y-intercept of the graph of an equation is the point , where b is the value of y when
. For the given equation,
when
. It follows that the y-intercept of the graph of the given equation is
. Additionally, for the given equation, the value of y doubles for each increase of 1 in the value of x. Therefore, the graph contains the points
,
,
, and
. Only the graph shown in choice D passes through these points.
Choices A and B are incorrect because these are graphs of decreasing, not increasing, exponential functions. Choice C is incorrect because the value of y increases by a growth factor greater than 2 for each increase of 1 in the value of x.
What is the x-intercept of the graph shown?
Choice B is correct. An x-intercept of a graph in the xy-plane is a point at which the graph crosses the x-axis. The graph shown crosses the x-axis at the point . Therefore, the x-intercept of the graph shown is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The expression can be rewritten as
, where k is a positive constant. What is the value of k ?
2
6
Choice D is correct. Factoring out the coefficient , the given expression can be rewritten as
. The expression
can be approached as a difference of squares and rewritten as
. Therefore, k must be
.
Choice A is incorrect. If k were 2, then the expression given would be rewritten as , which is equivalent to
, not
.
Choice B is incorrect. This may result from incorrectly factoring the expression and finding as the factored form of the expression. Choice C is incorrect. This may result from incorrectly distributing the
and rewriting the expression as
.
If , what is
?
–5
–2
2
5
Choice A is correct. Substituting –1 for x in the equation that defines f gives . Simplifying the expressions in the numerator and denominator yields
, which is equal to
or –5.
Choices B, C, and D are incorrect and may result from misapplying the order of operations when substituting –1 for x.
How many distinct real solutions does the given equation have?
Exactly one
Exactly two
Infinitely many
Zero
Choice D is correct. The number of solutions of a quadratic equation of the form , where , , and are constants, can be determined by the value of the discriminant, . If the value of the discriminant is positive, then the quadratic equation has exactly two distinct real solutions. If the value of the discriminant is equal to zero, then the quadratic equation has exactly one real solution. If the value of the discriminant is negative, then the quadratic equation has zero real solutions. In the given equation, , , , and . Substituting these values for , , and in yields , or . Since the value of its discriminant is negative, the given equation has zero real solutions. Therefore, the number of distinct real solutions the given equation has is zero.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The function gives a metal ball's height above the ground , in inches, seconds after it started moving on a track, where . Which of the following is the best interpretation of the vertex of the graph of in the xy-plane?
The metal ball's minimum height was inches above the ground.
The metal ball's minimum height was inches above the ground.
The metal ball's height was inches above the ground when it started moving.
The metal ball's height was inches above the ground when it started moving.
Choice A is correct. The graph of a quadratic equation in the form , where , , and are positive constants, is a parabola that opens upward with vertex . The given function is in the form , where , , , and . Therefore, the graph of is a parabola that opens upward with vertex . Since the parabola opens upward, the vertex is the lowest point on the graph. It follows that the y-coordinate of the vertex of the graph of is the minimum value of . Therefore, the minimum value of is . It’s given that represents the metal ball’s height above the ground, in inches, seconds after it started moving on a track. Therefore, the best interpretation of the vertex of the graph of is that the metal ball’s minimum height was inches above the ground.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The area of a triangle is square centimeters. The length of the base of the triangle is centimeters greater than the height of the triangle. What is the height, in centimeters, of the triangle?
Choice B is correct. The area, , of a triangle is given by the formula , where represents the length of the base of the triangle and represents its height. It’s given that the area of a triangle is square centimeters and that the length of the base of this triangle is centimeters greater than the height of the triangle. Let represent the height, in centimeters, of the triangle. It follows that the length of the base of the triangle can be expressed as . Substituting for , for , and for in the formula yields , or . Multiplying both sides of this equation by yields . Applying the distributive property on the right-hand side of this equation yields . Subtracting from both sides of this equation yields . In factored form, this equation is equivalent to . Applying the zero product property, it follows that or . Subtracting from both sides of the equation yields . Adding to both sides of the equation yields . Since represents the height of the triangle, it must be positive. Therefore, the height, in centimeters, of the triangle is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
If and
are the two solutions to the system of equations above, what is the value of
?
Choice D is correct. The system of equations can be solved using the substitution method. Solving the second equation for y gives y = –x – 1. Substituting the expression –x – 1 for y into the first equation gives –x – 1 = x2 + 2x + 1. Adding x + 1 to both sides of the equation yields x2 + 3x + 2 = 0. The left-hand side of the equation can be factored by finding two numbers whose sum is 3 and whose product is 2, which gives (x + 2)(x + 1) = 0. Setting each factor equal to 0 yields x + 2 = 0 and x + 1 = 0, and solving for x yields x = –2 or x = –1. These values of x can be substituted for x in the equation y = –x – 1 to find the corresponding y-values: y = –(–2) – 1 = 2 – 1 = 1 and y = –(–1) – 1 = 1 – 1 = 0. It follows that (–2, 1) and (–1, 0) are the solutions to the given system of equations. Therefore, (x1, y1) = (–2, 1), (x2, y2) = (–1, 0), and y1 + y2 = 1 + 0 = 1.
Choice A is incorrect. The solutions to the system of equations are (x1, y1) = (–2, 1) and (x2, y2) = (–1, 0). Therefore, –3 is the sum of the x-coordinates of the solutions, not the sum of the y-coordinates of the solutions. Choices B and C are incorrect and may be the result of computation or substitution errors.
The function gives the area of a rectangle, , if its width is and its length is times its width. Which of the following is the best interpretation of ?
If the width of the rectangle is , then the area of the rectangle is .
If the width of the rectangle is , then the length of the rectangle is .
If the width of the rectangle is , then the length of the rectangle is .
If the width of the rectangle is , then the area of the rectangle is .
Choice A is correct. The function gives the area of the rectangle, in , if its width is . Since the value of is the value of if , it follows that means that is if . In the given context, this means that if the width of the rectangle is , then the area of the rectangle is .
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from interpreting as the width, in , of the rectangle if its area is , rather than as the area, in , of the rectangle if its width is .
The expression is equivalent to
for some constant a. What is the value of a ?
2
3
4
7
Choice D is correct. It’s given that is equivalent to
for some constant a. Distributing the x over each term in the parentheses gives
, which is in the same form as the first given expression,
. The coefficient of the second term in
is 7. Therefore, the value of a is 7.
Choice A is incorrect. If the value of a were 2, then would be equivalent to
, which isn’t equivalent to
. Choice B is incorrect. If the value of a were 3, then
would be equivalent to
, which isn’t equivalent to
. Choice C is incorrect. If the value of a were 4, then
would be equivalent to
, which isn’t equivalent to
.
For the quadratic function , the table shows three values of and their corresponding values of . Which equation defines ?
Choice D is correct. The equation of a quadratic function can be written in the form , where , , and are constants. It’s given in the table that when , the corresponding value of is . Substituting for and for in the equation gives , which is equivalent to , or . It’s given in the table that when , the corresponding value of is . Substituting for and for in the equation gives , or . It’s given in the table that when , the corresponding value of is . Substituting for and for in the equation gives , which is equivalent to , or . Adding to the equation gives . Dividing both sides of this equation by gives . Since , substituting for into the equation gives . Subtracting from both sides of this equation gives . Substituting for in the equations and gives and , respectively. Since , substituting for in the equation gives , or . Subtracting from both sides of this equation gives . Dividing both sides of this equation by gives . Substituting for into the equation gives , or . Subtracting from both sides of this equation gives . Substituting for , for , and for in the equation gives , which is equivalent to , or . Therefore, defines .
Choice A is incorrect. If , then when , the corresponding value of is , not .
Choice B is incorrect. If , then when , the corresponding value of is , not .
Choice C is incorrect. If , then when , the corresponding value of is , not , and when , the corresponding value of is , not .
The given function models the number of coupons a company sent to their customers at the end of each year, where represents the number of years since the end of , and . If is graphed in the ty-plane, which of the following is the best interpretation of the y-intercept of the graph in this context?
The minimum estimated number of coupons the company sent to their customers during the years was .
The minimum estimated number of coupons the company sent to their customers during the years was .
The estimated number of coupons the company sent to their customers at the end of was .
The estimated number of coupons the company sent to their customers at the end of was .
Choice D is correct. The y-intercept of a graph in the ty-plane is the point where . For the given function , the y-intercept of the graph of in the ty-plane can be found by substituting for in the equation , which gives . This is equivalent to , or . Therefore, the y-intercept of the graph of is . It’s given that the function models the number of coupons a company sent to their customers at the end of each year. Therefore, represents the estimated number of coupons the company sent to their customers at the end of each year. It's also given that represents the number of years since the end of . Therefore, represents years since the end of , or the end of . Thus, the best interpretation of the y-intercept of the graph of is that the estimated number of coupons the company sent to their customers at the end of was .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The graph of a system of a linear equation and a nonlinear equation is shown. What is the solution to this system?
Choice C is correct. The solution to the system of two equations corresponds to the point where the graphs of the equations intersect. The graphs of the linear equation and the nonlinear equation shown intersect at the point . Thus, the solution to the system is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
If , what is t in terms of u ?
Choice D is correct. Multiplying both sides of the given equation by yields
. Dividing both sides of this equation by
yields
. Adding 2 to both sides of this equation yields
, which can be rewritten as
. Since the fractions on the right-hand side of this equation have a common denominator, adding the fractions yields
. Applying the distributive property to the numerator on the right-hand side of this equation yields
, which is equivalent to
.
Choices A, B, and C are incorrect and may result from various misconceptions or miscalculations.
In the given equation, is a constant. The equation has exactly one solution. What is the value of ?
Choice C is correct. It's given that the equation has exactly one solution. A quadratic equation of the form has exactly one solution if and only if its discriminant, , is equal to zero. It follows that for the given equation, and . Substituting for and for into yields , or . Since the discriminant must equal zero, . Subtracting from both sides of this equation yields . Dividing each side of this equation by yields . Therefore, the value of is .
Choice A is incorrect. If the value of is , this would yield a discriminant that is greater than zero. Therefore, the given equation would have two solutions, rather than exactly one solution.
Choice B is incorrect. If the value of is , this would yield a discriminant that is greater than zero. Therefore, the given equation would have two solutions, rather than exactly one solution.
Choice D is incorrect. If the value of is , this would yield a discriminant that is less than zero. Therefore, the given equation would have no real solutions, rather than exactly one solution.
In the given equation, is a positive constant. Which of the following is one of the solutions to the given equation?
Choice D is correct. If , then neither side of the given equation is defined and there can be no solution. Therefore, . Subtracting from both sides of the given equation yields , or . Squaring both sides of this equation yields , or . Since is positive and, therefore, nonzero, the expression is defined and equivalent to . It follows that the equation can be rewritten as , or , which is equivalent to . Adding to both sides of this equation yields . Taking the square root of both sides of this equation yields two solutions: and . Therefore, of the given choices, is one of the solutions to the given equation.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Which of the following functions has(have) a minimum value at ?
I only
II only
I and II
Neither I nor II
Choice D is correct. A function of the form , where and , is a decreasing function. Both of the given functions are of this form; therefore, both are decreasing functions. If a function is decreasing as the value of increases, the corresponding value of decreases; therefore, the function doesn’t have a minimum value. Thus, neither of the given functions has a minimum value.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
What is one of the solutions to the given equation?
Choice B is correct. Applying the zero product property to the given equation yields , , and . Dividing each side of the equation by yields . Adding to each side of the equation yields . Subtracting from each side of the equation yields . Therefore, the solutions to the given equation are , , and . Thus, one of the solutions to the given equation is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Which of the following is a solution to the given equation?
Choice D is correct. Adding to both sides of the given equation yields . To complete the square, adding , or , to both sides of this equation yields , or . Taking the square root of both sides of this equation yields , or . Subtracting from both sides of this equation yields . Therefore, the solutions to the given equation are and . Of these two solutions, only is given as a choice.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
For the function f defined above, what is the value of ?
9
12
18
36
Choice C is correct. The value of is found by evaluating the expression
when
. Substituting 2 for x in the given equation yields
. Simplifying
in the equation results in
. Evaluating the right-hand side of the equation yields
. Therefore, the value of
is 18.
Choice A is incorrect and may result from evaluating the expression as . Choice B is incorrect and may result from evaluating the expression as
. Choice D is incorrect and may result from evaluating the expression as
.
An object’s kinetic energy, in joules, is equal to the product of one-half the object’s mass, in kilograms, and the square of the object’s speed, in meters per second. What is the speed, in meters per second, of an object with a mass of 4 kilograms and kinetic energy of 18 joules?
3
6
9
36
Choice A is correct. It’s given that an object’s kinetic energy, in joules, is equal to the product of one-half the object’s mass, in kilograms, and the square of the object’s speed, in meters per second. This relationship can be represented by the equation , where K is the kinetic energy, m is the mass, and v is the speed. Substituting a mass of 4 kilograms for m and a kinetic energy of 18 joules for K results in the equation
, or
. Dividing both sides of this equation by 2 yields
. Taking the square root of both sides yields
and
. Since speed can’t be expressed as a negative number, the speed of the object is 3 meters per second.
Choice B is incorrect and may result from computation errors. Choice C is incorrect. This is the value of rather than v. Choice D is incorrect. This is the value of
rather than v.
The graph of a system of an absolute value function and a linear function is shown. What is the solution to this system of two equations?
Choice C is correct. The solution to the system of two equations corresponds to the point where the graphs of the equations intersect. The graphs of the linear function and the absolute value function shown intersect at the point . Thus, the solution to the system is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the y-intercept of the graph of the linear function.
Choice D is incorrect. This is the vertex of the graph of the absolute value function.
In the xy-plane, a parabola has vertex and intersects the x-axis at two points. If the equation of the parabola is written in the form , where , , and are constants, which of the following could be the value of ?
Choice D is correct. The equation of a parabola in the xy-plane can be written in the form , where is a constant and is the vertex of the parabola. If is positive, the parabola will open upward, and if is negative, the parabola will open downward. It’s given that the parabola has vertex . Substituting for and for in the equation gives , which can be rewritten as , or . Distributing the factor of on the right-hand side of this equation yields . Therefore, the equation of the parabola, , can be written in the form , where , , and . Substituting for and for in the expression yields , or . Since the vertex of the parabola, , is below the x-axis, and it’s given that the parabola intersects the x-axis at two points, the parabola must open upward. Therefore, the constant must have a positive value. Setting the expression equal to the value in choice D yields . Adding to both sides of this equation yields . Dividing both sides of this equation by yields , which is a positive value. Therefore, if the equation of the parabola is written in the form , where , , and are constants, the value of could be.
Choice A is incorrect. If the equation of a parabola with a vertex at is written in the form , where , , and are constants and , then the value of will be negative, which means the parabola will open downward, not upward, and will intersect the x-axis at zero points, not two points.
Choice B is incorrect. If the equation of a parabola with a vertex at is written in the form , where , , and are constants and , then the value of will be negative, which means the parabola will open downward, not upward, and will intersect the x-axis at zero points, not two points.
Choice C is incorrect. If the equation of a parabola with a vertex at is written in the form , where , , and are constants and , then the value of will be , which is inconsistent with the equation of a parabola.
What is the y-intercept of the graph shown?
Choice D is correct. The y-intercept of a graph in the xy-plane is the point at which the graph crosses the y-axis. The graph shown crosses the y-axis at the point . Therefore, the y-intercept of the graph shown is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The given equation relates the distinct positive numbers , , and . Which equation correctly expresses in terms of and ?
Choice D is correct. To express in terms of and , the given equation must be solved for . Dividing each side of the given equation by yields .
Choice A is incorrect. This is equivalent to .
Choice B is incorrect. This is equivalent to .
Choice C is incorrect. This is equivalent to .
The parabola shown intersects the y-axis at the point . What is the value of ?
The correct answer is . It's given that the parabola intersects the y-axis at the point . The graph shows that the parabola intersects the y-axis at the point . Therefore, the value of is .
The graphs of the equations in the given system of equations intersect at the point in the -plane. What is a possible value of ?
Choice A is correct. It's given that the graphs of the equations in the given system of equations intersect at the point . Therefore, this intersection point is a solution to the given system. The solution can be found by isolating in each equation. The given equation can be rewritten to isolate by subtracting from both sides of the equation, which gives . The given equation can be rewritten to isolate by subtracting from both sides of the equation, which gives . With each equation solved for , the value of from one equation can be substituted into the other, which gives . Adding and to both sides of this equation results in . Dividing both sides of this equation by results in . This equation can be rewritten by factoring the left-hand side, which yields . By the zero-product property, if , then , or . It follows that , or . Since only is given as a choice, a possible value of is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
| x | p(x) |
The table above gives selected values of a polynomial function p. Based on the values in the table, which of the following must be a factor of p ?
Choice D is correct. According to the table, when x is or 2,
. Therefore, two x-intercepts of the graph of p are
and
. Since
and
are x-intercepts, it follows that
and
are factors of the polynomial equation. This is because when
, the value of
is 0. Similarly, when
, the value of
is 0. Therefore, the product
is a factor of the polynomial function p.
Choice A is incorrect. The factor corresponds to an x-intercept of
, which isn’t present in the table. Choice B is incorrect. The factor
corresponds to an x-intercept of
, which isn’t present in the table. Choice C is incorrect. The factors
and
correspond to x-intercepts
and
, respectively, which aren’t present in the table.
A ball is dropped from an initial height of feet and bounces off the ground repeatedly. The function estimates that the maximum height reached after each time the ball hits the ground is of the maximum height reached after the previous time the ball hit the ground. Which equation defines , where is the estimated maximum height of the ball after it has hit the ground times and is a whole number greater than and less than ?
Choice B is correct. It's given that for the function , is the estimated maximum height, in feet, of the ball after it has hit the ground times. It's also given that the function estimates that the maximum height reached after each time the ball hits the ground is of the maximum height reached after the previous time the ball hit the ground. It follows that is a decreasing exponential function that can be written in the form , where is the initial height, in feet, the ball was dropped from and the function estimates that the maximum height reached after each time the ball hits the ground is of the maximum height reached after the previous time the ball hit the ground. It's given that the ball is dropped from an initial height of feet. Therefore, . Since the function estimates that the maximum height reached after each time the ball hits the ground is of the maximum height reached after the previous time the ball hit the ground, . Substituting for and for in the equation yields , or .
Choice A is incorrect. This function estimates that the maximum height reached after each time the ball hits the ground is , not , of the maximum height reached after the previous time the ball hit the ground.
Choice C is incorrect. This function estimates that the ball is dropped from an initial height of feet, not feet, and that the maximum height reached after each time the ball hits the ground is , not , of the maximum height reached after the previous time the ball hit the ground.
Choice D is incorrect. This function estimates that the ball is dropped from an initial height of feet, not feet.
What is the y-intercept of the graph shown?
Choice A is correct. The y-intercept of a graph in the xy-plane is the point on the graph where . For the graph shown, at , the corresponding value of y is . Therefore, the y-intercept of the graph shown is .
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
The height, in feet, of an object x seconds after it is thrown straight up in the air can be modeled by the function . Based on the model, which of the following statements best interprets the equation
?
The height of the object 1.4 seconds after being thrown straight up in the air is 1.64 feet.
The height of the object 1.64 seconds after being thrown straight up in the air is 1.4 feet.
The height of the object 1.64 seconds after being thrown straight up in the air is approximately 1.4 times as great as its initial height.
The speed of the object 1.4 seconds after being thrown straight up in the air is approximately 1.64 feet per second.
Choice A is correct. The value 1.4 is the value of x, which represents the number of seconds after the object was thrown straight up in the air. When the function h is evaluated for x = 1.4, the function has a value of 1.64, which is the height, in feet, of the object.
Choices B and C are incorrect and may result from misidentifying seconds as feet and feet as seconds. Additionally, choice C may result from incorrectly including the initial height of the object as the input x. Choice D is incorrect and may result from misidentifying height as speed.
The speed of sound in dry air, v, can be modeled by the formula , where T is the temperature in degrees Celsius and v is measured in meters per second. Which of the following correctly expresses T in terms of v ?
Choice D is correct. To express T in terms of v, subtract 331.3 from both sides of the equation, which gives v – 331.3 = 0.606T. Dividing both sides of the equation by 0.606 gives .
Choices A, B, and C are incorrect and are the result of incorrect steps while solving for T.
The function is defined by . What is the value of ?
Choice C is correct. It's given that . Substituting for in this equation yields . This is equivalent to , or .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect. This is the value of , not .
Choice D is incorrect. This is the value of , not .
Which expression is equivalent to ?
Choice A is correct. Applying the distributive property, the given expression can be written as . Grouping like terms in this expression yields . Combining like terms in this expression yields .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
What is the sum of the solutions to the given equation?
The correct answer is . Applying the distributive property to the left-hand side of the given equation, , yields . Applying the distributive property to the right-hand side of the given equation, , yields . Thus, the equation becomes . Combining like terms on the left- and right-hand sides of this equation yields , or . For a quadratic equation in the form , where , , and are constants, the quadratic formula gives the solutions to the equation in the form . Substituting for , for , and for from the equation into the quadratic formula yields , or . It follows that the solutions to the given equation are and . Adding these two solutions gives the sum of the solutions: , which is equivalent to , or . Note that 29/3, 9.666, and 9.667 are examples of ways to enter a correct answer.
During the first part of an experiment, a ball was launched from a -foot-tall platform. The graph shows the height , in feet, of the ball seconds after it was launched during the first part of the experiment.
During the second part of the experiment, the ball was launched the same way, but from a platform that is feet shorter than the first platform. Which of the following graphs could represent the height , in feet, of the ball seconds after it was launched during the second part of the experiment?
Choice B is correct. It's given that represents the height, in feet, of the ball seconds after it was launched. It's also given that during the first part of an experiment, a ball was launched from a -foot-tall platform. Therefore, the y-coordinate of the y-intercept of the given graph, , represents the platform height, in feet. During the second part of the experiment, the platform the ball was launched from was feet shorter than the platform in the first part of the experiment. It follows that the height of the platform in the second part of the experiment was feet, or feet. Therefore, the y-coordinate of the y-intercept of the graph representing the second part of the experiment must be . Only choice B satisfies this condition.
Choice A is incorrect. This could represent the graph if the ball were launched from a platform that was about feet shorter rather than feet shorter.
Choice C is incorrect. This could represent the graph if the ball were launched from a platform that was feet taller rather than feet shorter.
Choice D is incorrect. This could represent the graph if the ball were launched from a platform that was twice as tall rather than feet shorter.
When the quadratic function is graphed in the xy-plane, where , its vertex is . One of the x-intercepts of this graph is . What is the other x-intercept of the graph?
Choice B is correct. Since the line of symmetry for the graph of a quadratic function contains the vertex of the graph, the x-coordinate of the vertex of the graph of is the x-coordinate of the midpoint of its two x-intercepts. The midpoint of two points with x-coordinates and has x-coordinate , where . It′s given that the vertex is and one of the x-intercepts is . Substituting for and for in the equation yields . Multiplying each side of this equation by yields . Adding to each side of this equation yields . Therefore, the other x-intercept is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Function is a quadratic function where and . The graph of in the xy-plane has a vertex at . What is the value of ?
The correct answer is . It’s given that function is a quadratic function where and . It follows that the graph of in the xy-plane passes through the points and . When the graph of a quadratic function contains two points and , the x-coordinate of the vertex of the graph is the average of and . Therefore, the x-coordinate of the vertex of the graph of is , or . It's given that the graph of in the xy-plane has a vertex at . It follows that the value of is .
The function f is defined by . The graph of f in the xy-plane is a parabola. Which of the following intervals contains the x-coordinate of the vertex of the graph of f ?
Choice B is correct. The graph of a quadratic function in the xy-plane is a parabola. The axis of symmetry of the parabola passes through the vertex of the parabola. Therefore, the vertex of the parabola and the midpoint of the segment between the two x-intercepts of the graph have the same x-coordinate. Since , the x-coordinate of the vertex is
. Of the shown intervals, only the interval in choice B contains –2. Choices A, C, and D are incorrect and may result from either calculation errors or misidentification of the graph’s x-intercepts.
If , what is the value of ?
The correct answer is . An expression of the form , where and are integers greater than and , is equivalent to . Therefore, the expression on the right-hand side of the given equation, , is equivalent to . Thus, . It follows that . Dividing both sides of this equation by yields . Note that 7/24, .2916, .2917, 0.219, and 0.292 are examples of ways to enter a correct answer.
If is a solution to the system of equations above, which of the following is a possible value of x?
0
1
2
3
Choice D is correct. Substituting from the second equation for y in the first equation yields
. Subtracting 12 from both sides of this equation and rewriting the equation results in
. Factoring the left-hand side of this equation yields
. Using the zero product property to solve for x, it follows that
and
. Solving each equation for x yields
and
, respectively. Thus, two possible values of x are 3 and
. Of the choices given, 3 is the only possible value of x.
Choices A, B, and C are incorrect. Substituting 0 for x in the first equation gives , or
; then, substituting 12 for y and 0 for x in the second equation gives
, or
, which is false. Similarly, substituting 1 or 2 for x in the first equation yields
or
, respectively; then, substituting 11 or 10 for y in the second equation yields a false statement.
In the given system of equations, is a positive integer constant. The system has no real solutions. What is the least possible value of ?
The correct answer is . It's given by the first equation of the system of equations that . Substituting for in the second given equation, , yields . Adding to both sides of this equation yields . A quadratic equation of the form , where , , and are constants, has no real solutions if and only if its discriminant, , is negative. In the equation , where is a positive integer constant, , , and . Substituting for , for , and for in yields , or . Since this value must be negative, . Adding to both sides of this inequality yields . Dividing both sides of this inequality by yields . Subtracting from both sides of this inequality yields . Since is a positive integer constant, the least possible value of is .
The function is defined by , where and are constants and . In the xy-plane, the graph of has a y-intercept at and passes through the point . What is the value of ?
The correct answer is . It's given that function is defined by , where and are constants and . It's also given that the graph of in the xy-plane has a y-intercept at and passes through the point . Since the graph has a y-intercept at , . Substituting for in the given equation yields , or , and substituting for in this equation yields . Subtracting from each side of this equation yields . Substituting for in the equation yields . Since the graph also passes through the point , . Substituting for in the equation yields , and substituting for yields . Adding to each side of this equation yields . Taking the square root of both sides of this equation yields . Since it's given that , the value of is . It follows that the value of is , or .
The graph shows a marble's height above the ground , in inches, seconds after it started moving on an elevated track of a marble run. Which of the following is the best interpretation of the -intercept of the graph?
The marble's height was inches above the ground seconds after it started moving.
The marble's height was inches above the ground when it started moving.
The marble's minimum height was inches above the ground.
The marble's minimum height was inches above the ground.
Choice B is correct. The y-intercept of a graph is the point at which the graph intersects the y-axis. The graph shown intersects the y-axis at the point . Therefore, the y-intercept of the graph is . It’s given that is the height of the marble above the ground, in inches, and is the number of seconds after the marble started moving. It follows that the marble's height was inches above the ground seconds after it started moving. Therefore, the best interpretation of the y-intercept of the graph is that the marble’s height was inches above the ground when it started moving.
Choice A is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
An engineer wanted to identify the best angle for a cooling fan in an engine in order to get the greatest airflow. The engineer discovered that the function above models the airflow , in cubic feet per minute, as a function of the angle of the fan
, in degrees. According to the model, what angle, in degrees, gives the greatest airflow?
–0.28
0.28
27
880
Choice C is correct. The function f is quadratic, so it will have either a maximum or a minimum at the vertex of the graph. Since the coefficient of the quadratic term (–0.28) is negative, the vertex will be at a maximum. The equation f() = –0.28(
– 27)2 + 880 is given in vertex form, so the vertex is at
= 27. Therefore, an angle of 27 degrees gives the greatest airflow.
Choices A and B are incorrect and may be the result of misidentifying which value in a quadratic equation in vertex form represents the vertex. Choice D is incorrect. This choice identifies the maximum value of f() rather than the value of
for which f(
) is maximized.
The given equation relates the numbers and , where is not equal to and . Which equation correctly expresses in terms of ?
Choice D is correct. To express in terms of , the given equation must be solved for . Subtracting from both sides of the given equation yields . Since is not equal to , multiplying both sides of this equation by yields . It's given that , which means is not equal to . Therefore, dividing both sides of by yields , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The function gives the number of bacteria in a population minutes after an initial observation. How much time, in minutes, does it take for the number of bacteria in the population to double?
The correct answer is . It's given that minutes after an initial observation, the number of bacteria in a population is . This expression consists of the initial number of bacteria, , multiplied by the expression . The time it takes for the number of bacteria to double is the increase in the value of that causes the expression to double. Since the base of the expression is , the expression will double when the exponent increases by . Since the exponent of the expression is , the exponent will increase by when increases by . Therefore the time, in minutes, it takes for the number of bacteria in the population to double is .
Which ordered pair is the solution to the given system of equations?
Choice C is correct. The second equation in the given system of equations is . Substituting for in the first equation of the given system yields . Factoring out of the left-hand side of this equation yields . An expression with a factor of the form is equal to zero when . Each side of this equation has a factor of , so each side of the equation is equal to zero when . Substituting for into the equation yields , or , which is true. Substituting for into the second equation in the given system of equations yields , or . Therefore, the solution to the system of equations is the ordered pair .
Choice A is incorrect and may result from switching the order of the solutions for and .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Which expression is equivalent to ?
Choice A is correct. The expression can be rewritten as , which is equivalent to .
Choice B is incorrect. This expression is equivalent to , not .
Choice C is incorrect. This expression is equivalent to , not .
Choice D is incorrect. This expression is equivalent to , not .
Which expression is equivalent to ?
Choice C is correct. Since each term of the given expression has a common factor of , it may be rewritten as , or .
Choice A is incorrect. This expression is equivalent to , not .
Choice B is incorrect. This expression is equivalent to , not .
Choice D is incorrect. This expression is equivalent to , not .
What is the sum of the solutions to the given equation?
The correct answer is . Subtracting from each side of the given equation yields . By the definition of absolute value, if , then or . Adding to each side of the equation yields . Adding to each side of the equation yields . Therefore, the solutions to the given equation are and , and it follows that the sum of the solutions to the given equation is , or .
The given equation relates the positive numbers , , and . Which equation correctly expresses in terms of and ?
Choice C is correct. Multiplying each side of the given equation by yields the equivalent equation . Dividing each side of this equation by yields , or .
Choice A is incorrect. This equation is not equivalent to the given equation.
Choice B is incorrect. This equation is not equivalent to the given equation.
Choice D is incorrect. This equation is not equivalent to the given equation.
Which of the following is a solution to the given equation?
Choice C is correct. Since a product of two factors is equal to if and only if at least one of the factors is , either or . Subtracting from each side of the equation yields . Dividing each side of this equation by yields . Adding to each side of the equation yields . Dividing each side of this equation by yields . It follows that the solutions to the given equation are and . Therefore, is a solution to the given equation.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
| Time (years) | Total amount (dollars) |
|---|---|
Sara opened a savings account at a bank. The table shows the exponential relationship between the time , in years, since Sara opened the account and the total amount , in dollars, in the account. If Sara made no additional deposits or withdrawals, which of the following equations best represents the relationship between and ?
Choice B is correct. It’s given that the relationship between and is exponential. The table shows that the value of increases as the value of increases. Therefore, the relationship between and can be represented by an increasing exponential equation of the form , where and are positive constants. The table shows that when , . Substituting for and for in the equation yields , which is equivalent to , or . Substituting for in the equation yields . The table also shows that when , . Substituting for and for in the equation yields , or . Dividing both sides of this equation by yields . Subtracting from both sides of this equation yields . Substituting for in the equation yields . Therefore, of the choices, choice B best represents the relationship between and .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Which value is a solution to the given equation?
Choice B is correct. Subtracting from each side of the given equation yields . By the definition of absolute value, if , then or . Of the given choices, is a solution to the given equation.
Choice A is incorrect. This is the quotient, not the difference, of and .
Choice C is incorrect. This is the sum, not the difference, of and .
Choice D is incorrect and may result from conceptual or calculation errors.
A landscaper is designing a rectangular garden. The length of the garden is to be 5 feet longer than the width. If the area of the garden will be 104 square feet, what will be the length, in feet, of the garden?
The correct answer is 13. Let w represent the width of the rectangular garden, in feet. Since the length of the garden will be 5 feet longer than the width of the garden, the length of the garden will be feet. Thus the area of the garden will be
. It is also given that the area of the garden will be 104 square feet. Therefore,
, which is equivalent to
. Factoring this equation results in
. Therefore,
and
. Because width cannot be negative, the width of the garden must be 8 feet. This means the length of the garden must be
feet.
The function models the population, in thousands, of a certain city years after . According to the model, the population is predicted to increase by every months. What is the value of ?
Choice C is correct. It's given that the function models the population of the city years after . Since there are months in a year, months is equivalent to years. Therefore, the expression can represent the number of years in -month periods. Substituting for in the given equation yields , which is equivalent to . Therefore, for each -month period, the predicted population of the city is times, or of, the previous population. This means that the population is predicted to increase by every months.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. Each year, the predicted population of the city is times the previous year's predicted population, which is not the same as an increase of .
Choice D is incorrect and may result from conceptual or calculation errors.
The graph of a system of a linear and a quadratic equation is shown. What is the solution to this system?
Choice D is correct. The solution to the system corresponds to the point where the graphs of the equations intersect. The graphs of the linear equation and the quadratic equation shown intersect at the point . Therefore, is the solution to this system.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Which of the following is equivalent to ?
5x
5x2
Choice A is correct. Since is a common term in the original expression, like terms can be added:
. Distributing the constant term 5 yields
.
Choice B is incorrect and may result from not distributing the negative signs in the expressions within the parentheses. Choice C is incorrect and may result from not distributing the negative signs in the expressions within the parentheses and from incorrectly eliminating the -term. Choice D is incorrect and may result from incorrectly eliminating the x-term.
The given equation relates the positive numbers , , and . Which equation correctly gives in terms of and ?
Choice A is correct. Dividing each side of the given equation by yields .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This equation is equivalent to , not .
Choice D is incorrect and may result from conceptual or calculation errors.
The function is defined by . What is the value of ?
The correct answer is . It's given that the function is defined by . Substituting for in this equation yields , or , which is equivalent to . Therefore, the value of is .
Which expression is equivalent to ?
Choice B is correct. Since each term in the given expression has a common factor of , it can be rewritten as , or . Therefore, the expression is equivalent to .
Alternate approach: Since the two terms of the given expression are both constant multiples of , they are like terms and can, therefore, be combined through subtraction. Subtracting like terms in the expression yields .
Choice A is incorrect. This expression is equivalent to , not .
Choice C is incorrect. This expression is equivalent to , not .
Choice D is incorrect and may result from conceptual or calculation errors.
In the xy-plane, the graph of intersects line p at
and
, where a and b are constants. What is the slope of line p ?
6
2
Choice A is correct. It’s given that the graph of and line p intersect at
and
. Therefore, the value of y when
is the value of a, and the value of y when
is the value of b. Substituting 1 for x in the given equation yields
, or
. Similarly, substituting 5 for x in the given equation yields
, or
. Therefore, the intersection points are
and
. The slope of line p is the ratio of the change in y to the change in x between these two points:
, or 6.
Choices B, C, and D are incorrect and may result from conceptual or calculation errors in determining the values of a, b, or the slope of line p.
Which expression is equivalent to , where ?
Choice B is correct. Since , multiplying the exponent of the given expression by yields an equivalent expression: . Since , the expression can be rewritten as . Applying properties of exponents, this expression can be rewritten as . An expression of the form , where and , is equivalent to . Therefore, is equivalent to .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Which of the following is equivalent to ?
Choice D is correct. The expression has two terms,
and 4. The greatest common factor of these two terms is 2. Factoring 2 from each of these terms yields
, or
.
Choices A and B are incorrect because 4 is not a factor of the term . Choice C is incorrect and may result from factoring 2 from
but not from 4.
For the function , , and for each increase in by , the value of decreases by . What is the value of ?
The correct answer is . It’s given that and that for each increase in by , the value of decreases by . Because the output of the function decreases by a constant percentage for each -unit increase in the value of , this relationship can be represented by an exponential function of the form , where represents the initial value of the function and represents the rate of decay,
expressed as a decimal. Because , the value of must be . Because the value of decreases by for each -unit increase in , the value of must be , or . Therefore, the function can be defined by . Substituting for in this function yields , which is equivalent to , or . Either or may be entered as the correct answer.
Alternate approach: It’s given that and that for each increase in by , the value of decreases by . Therefore, when , the value of is , or , of , which can be expressed as . Since , the value of is . Similarly, when , the value of is of , which can be expressed as . Since , the value of is . Either or may be entered as the correct answer.
A competitive diver dives from a platform into the water. The graph shown gives the height above the water , in meters, of the diver seconds after diving from the platform. What is the best interpretation of the x-intercept of the graph?
The diver reaches a maximum height above the water at seconds.
The diver hits the water at seconds.
The diver reaches a maximum height above the water at seconds.
The diver hits the water at seconds.
Choice B is correct. It’s given that the graph shows the height above the water , in meters, of a diver seconds after diving from a platform. The x-intercept of a graph is the point at which the graph intersects the x-axis, or when the value of is . The graph shown intersects the x-axis between and . In other words, the diver is meters above the water, or hits the water, between and seconds after diving from the platform. Of the given choices, only choice B includes an interpretation where the diver hits the water between and seconds. Therefore, the best interpretation of the x-intercept of the graph is the diver hits the water at seconds.
Choice A is incorrect and may result from conceptual errors.
Choice C is incorrect. This is the best interpretation of the maximum value, not the x-intercept, of the graph.
Choice D is incorrect and may result from conceptual errors.
The function is defined by . What is the value of ?
The correct answer is . The value of is the value of when . Substituting for in the given equation yields , which is equivalent to , or . Therefore, the value of is . Note that 1/2 and .5 are examples of ways to enter a correct answer.
The equation above models the number of members, M, of a gym t years after the gym opens. Of the following, which equation models the number of members of the gym q quarter years after the gym opens?
Choice A is correct. In 1 year, there are 4 quarter years, so the number of quarter years, q, is 4 times the number of years, t ; that is, . This is equivalent to
, and substituting this into the expression for M in terms of t gives
.
Choices B and D are incorrect and may be the result of incorrectly using instead of
. (Choices B and D are nearly the same since
is equivalent to
, which is approximately
.) Choice C is incorrect and may be the result of incorrectly using
and unnecessarily dividing 0.02 by 4.
A certain college had 3,000 students enrolled in 2015. The college predicts that after 2015, the number of students enrolled each year will be 2% less than the number of students enrolled the year before. Which of the following functions models the relationship between the number of students enrolled, , and the number of years after 2015, x ?
Choice D is correct. Because the change in the number of students decreases by the same percentage each year, the relationship between the number of students and the number of years can be modeled with a decreasing exponential function in the form , where
is the number of students, a is the number of students in 2015, r is the rate of decrease each year, and x is the number of years since 2015. It’s given that 3,000 students were enrolled in 2015 and that the rate of decrease is predicted to be 2%, or 0.02. Substituting these values into the decreasing exponential function yields
, which is equivalent to
.
Choices A, B, and C are incorrect and may result from conceptual errors when translating the given information into a decreasing exponential function.
A solution to the given system of equations is . What is the greatest possible value of ?
Choice A is correct. It's given that and ; therefore, it follows that . This equation can be rewritten as . Subtracting from both sides of this equation yields . This equation can be rewritten as , or . By the zero product property, or . Subtracting from both sides of the equation yields . Subtracting from both sides of the equation yields . Therefore, the given system of equations has solutions, , that occur when and . Since is greater than , the greatest possible value of is .
Choice B is incorrect. This is the negative of the greatest possible value of when for the second equation in the given system of equations.
Choice C is incorrect. This is the value of when for the first equation in the given system of equations.
Choice D is incorrect. This is the value of when for the second equation in the given system of equations.
The function gives a company’s predicted annual revenue, in dollars, years after the company started selling light bulbs online, where . What is the best interpretation of the statement “ is approximately equal to ” in this context?
years after the company started selling light bulbs online, its predicted annual revenue is approximately dollars.
years after the company started selling light bulbs online, its predicted annual revenue will have increased by a total of approximately dollars.
When the company’s predicted annual revenue is approximately dollars, it is times the predicted annual revenue for the previous year.
When the company’s predicted annual revenue is approximately dollars, it is greater than the predicted annual revenue for the previous year.
Choice A is correct. It's given that the function gives a company's predicted annual revenue, in dollars, years after the company started selling light bulbs online. It follows that represents the company's predicted annual revenue, in dollars, years after the company started selling light bulbs online. Since the value of is the value of when , it follows that " is approximately equal to " means that is approximately equal to when . Therefore, the best interpretation of the statement " is approximately equal to " in this context is years after the company started selling light bulbs online, its predicted annual revenue is approximately dollars.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
What is the solution to the given equation?
Choice C is correct. Since the left-hand side of the given equation has a factor of in both the numerator and the denominator, the solution to the given equation can be found by solving the equation . Adding to both sides of this equation yields . Substituting for in the given equation yields , or . Therefore, the solution to the given equation is .
Choice A is incorrect. Substituting for in the given equation yields , or , which is false.
Choice B is incorrect. Substituting for in the given equation yields , or , which is false.
Choice D is incorrect. Substituting for in the given equation yields , or , which is false.
The given equation relates the positive variables , , , and . Which of the following is equivalent to ?
Choice C is correct. Multiplying each side of the given equation by yields . Distributing on each side of this equation yields , or . Adding to each side of this equation yields . Multiplying by , by , and by yields , which is equivalent to . Since , and it's given that , , , and are positive, it follows that the reciprocals of each side of this equation are also equal. Thus, , or . Therefore, is equivalent to .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The sum of and
can be written in the form
, where a, b, and c are constants. What is the value of
?
The correct answer is 32. The sum of the given expressions is . Combining like terms yields
. Based on the form of the given equation,
,
, and
. Therefore,
.
Alternate approach: Because is the value of
when
, it is possible to first make that substitution into each polynomial before adding them. When
, the first polynomial is equal to
and the second polynomial is equal to
. The sum of 30 and 2 is 32.
The expression above is equivalent to , where a, b, and c are constants. What is the value of b?
The correct answer is . The expression
can be written in the form
, where a, b, and c are constants, by multiplying out the expression using the distributive property of multiplication over addition. The result is
. This expression can be rewritten by multiplying as indicated to give
, which can be simplified to
, or
. This is in the form
, where the value of b is
. Note that 5/2 and 2.5 are examples of ways to enter a correct answer.
The functions and are defined by the given equations, where . Which of the following equations displays, as a constant or coefficient, the maximum value of the function it defines, where ?
I only
II only
I and II
Neither I nor II
Choice B is correct. Functions and are both exponential functions with a base of . Since is less than , functions and are both decreasing exponential functions. This means that and decrease as increases. Since and decrease as increases, the maximum value of each function occurs at the least value of for which the function is defined. It's given that functions and are defined for . Therefore, the maximum value of each function occurs at . Substituting for in the equation defining yields , which is equivalent to , or . Therefore, the maximum value of is . Since the equation doesn't display the value , the equation defining doesn't display the maximum value of . Substituting for in the equation defining yields , which can be rewritten as , or , which is equivalent to . Therefore, the maximum value of is . Since the equation displays the value , the equation defining displays the maximum value of . Thus, only equation II displays, as a constant or coefficient, the maximum value of the function it defines.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The function is defined by , where , , and are constants. The graph of in the xy-plane passes through the points and . If is an integer greater than , which of the following could be the value of ?
Choice A is correct. It's given that the graph of in the xy-plane passes through the points and . It follows that when the value of is either or , the value of is . It's also given that the function is defined by , where , , and are constants. It follows that the function is a quadratic function and, therefore, may be written in factored form as , where the value of is when is either or . Since the value of is when the value of is either or , and the value of is when the value of is either or , it follows that and are equal to and . Substituting for and for in the equation yields , or . Distributing the right-hand side of this equation yields , or . Since it's given that , it follows that . Adding to each side of this equation yields . Since , if is an integer, the value of must be a multiple of . If is an integer greater than , it follows that . Therefore, . It follows that the value of is less than or equal to , or . Of the given choices, only is a multiple of that's less than or equal to .
Choice B is incorrect. This is the value of if is equal to, not greater than, .
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Blood volume,, in a human can be determined using the equation
, where
is the plasma volume and H is the hematocrit (the fraction of blood volume that is red blood cells). Which of the following correctly expresses the hematocrit in terms of the blood volume and the plasma volume?
Choice A is correct. The hematocrit can be expressed in terms of the blood volume and the plasma volume by solving the given equation for H. Multiplying both sides of this equation by
yields
. Dividing both sides by
yields
. Subtracting 1 from both sides yields
. Dividing both sides by
yields
.
Choices B, C, and D are incorrect and may result from errors made when manipulating the equation.
What is the positive solution to the given equation?
Choice D is correct. Adding to each side of the given equation yields . Taking the square root of each side of this equation yields . Therefore, the positive solution to the given equation is .
Choice A is incorrect. This is the positive solution to the equation , not .
Choice B is incorrect. This is the positive solution to the equation , not .
Choice C is incorrect. This is the positive solution to the equation , not .
A model predicts that the population of Bergen was in . The model also predicts that each year for the next years, the population increased by of the previous year's population. Which equation best represents this model, where is the number of years after , for ?
Choice D is correct. It's given that a model predicts the population of Bergen in was . The model also predicts that each year for the next years, the population increased by of the previous year's population. The predicted population in one of these years can be found by multiplying the predicted population from the previous year by . Since the predicted population in was , the predicted population year later is . The predicted population years later is this value times , which is , or . The predicted population years later is this value times , or . More generally, the predicted population, , years after is represented by the equation .
Choice A is incorrect. Substituting for in this equation indicates the predicted population in was rather than .
Choice B is incorrect. Substituting for in this equation indicates the predicted population in was rather than .
Choice C is incorrect. This equation indicates the predicted population is decreasing, rather than increasing, by each year.
What is the solution to the given equation?
The correct answer is . Since is in the denominator of a fraction in the given equation, can't be equal to . Since isn't equal to , multiplying both sides of the given equation by yields an equivalent equation, . Dividing both sides of this equation by yields . Therefore, is the solution to the given equation.
The power P produced by a machine is represented by the equation above, where W is the work performed during an amount of time t. Which of the following correctly expresses W in terms of P and t ?
Choice A is correct. Multiplying both sides of the equation by t yields , or
, which expresses W in terms of P and t. This is equivalent to W = Pt.
Choices B, C, and D are incorrect. Each of the expressions given in these answer choices gives W in terms of P and t but doesn’t maintain the given relationship between W, P, and t. These expressions may result from performing different operations with t on each side of the equation. In choice B, W has been multiplied by t, and P has been divided by t. In choice C, W has been multiplied by t, and the quotient of P divided by t has been reciprocated. In choice D, W has been multiplied by t, and P has been added to t. However, in order to maintain the relationship between the variables in the given equation, the same operation must be performed with t on each side of the equation.
If one solution to the system of equations above is , what is one possible value of x ?
The correct answer is either 8 or 9. The first equation can be rewritten as . Substituting
for y in the second equation gives
. By applying the distributive property, this can be rewritten as
. Subtracting 72 from both sides of the equation yields
. Factoring the left-hand side of this equation yields
. Applying the Zero Product Property, it follows that
and
. Solving each equation for x yields
and
respectively. Note that 8 and 9 are examples of ways to enter a correct answer.
Which expression is equivalent to , where and ?
Choice A is correct. For positive values of , , where and are integers. Since it's given that and , this property can be applied to rewrite the given expression as , which is equivalent to . For positive values of , . This property can be applied to rewrite the expression as , which is equivalent to .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Which expression is equivalent to ?
Choice B is correct. Since is a common factor of each term in the given expression, the expression can be rewritten as .
Choice A is incorrect. This expression is equivalent to .
Choice C is incorrect. This expression is equivalent to .
Choice D is incorrect. This expression is equivalent to .
A rubber ball bounces upward one-half the height that it falls each time it hits the ground. If the ball was originally dropped from a distance of 20.0 feet above the ground, what was its maximum height above the ground, in feet, between the third and fourth time it hit the ground?
The correct answer is 2.5. After hitting the ground once, the ball bounces to feet. After hitting the ground a second time, the ball bounces to
feet. After hitting the ground for the third time, the ball bounces to
feet. Note that 2.5 and 5/2 are examples of ways to enter a correct answer.
The function is defined by . What is the value of ?
The correct answer is . The value of is the value of when . Substituting for in the given equation yields , which is equivalent to , or . Thus, the value of is .
The function gives a company’s predicted annual revenue, in dollars, years after the company started selling jewelry online, where . What is the best interpretation of the statement “ is approximately equal to ” in this context?
years after the company started selling jewelry online, its predicted annual revenue is approximately dollars.
years after the company started selling jewelry online, its predicted annual revenue will have increased by a total of approximately dollars.
When the company’s predicted annual revenue is approximately dollars, it is times the predicted annual revenue for the previous year.
When the company’s predicted annual revenue is approximately dollars, it is greater than the predicted annual revenue for the previous year.
Choice A is correct. It’s given that the function gives a company’s predicted annual revenue, in dollars, years after the company started selling jewelry online. Since the value of is the value of when , it follows that “ is approximately equal to ” means that is approximately equal to when . Therefore, the best interpretation of the given statement is that years after the company started selling jewelry online, its predicted annual revenue is approximately dollars.
Choice B is incorrect. The function gives the predicted annual revenue, not the predicted increase in annual revenue.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect. In the given function, represents the number of years after the company started selling jewelry online, not the percent increase in revenue from the previous year.
Which expression is equivalent to ?
Choice A is correct. Since is a factor of each term in the given expression, the expression is equivalent to , or .
Choice B is incorrect. This expression is equivalent to , not .
Choice C is incorrect. This expression is equivalent to , not .
Choice D is incorrect. This expression is equivalent to , not .
The formula above can be used to approximate the dew point D, in degrees Fahrenheit, given the temperature T, in degrees Fahrenheit, and the relative humidity of H percent, where . Which of the following expresses the relative humidity in terms of the temperature and the dew point?
Choice A is correct. It’s given that . Solving this formula for H expresses the relative humidity in terms of the temperature and the dew point. Subtracting T from both sides of this equation yields
. Multiplying both sides by
yields
. Subtracting 100 from both sides yields
. Multiplying both sides by
results in the formula
.
Choices B, C, and D are incorrect and may result from errors made when rewriting the given formula.
The number of bacteria in a liquid medium doubles every day. There are bacteria in the liquid medium at the start of an observation. Which represents the number of bacteria, , in the liquid medium days after the start of the observation?
Choice D is correct. Since the number of bacteria doubles every day, the relationship between and can be represented by an exponential equation of the form , where is the number of bacteria at the start of the observation and the number of bacteria increases by a factor of every day. It’s given that there are bacteria at the start of the observation. Therefore, . It’s also given that the number of bacteria doubles, or increases by a factor of , every day. Therefore, . Substituting for and for in the equation yields .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This equation represents a situation where the number of bacteria is decreasing by half, not doubling, every day.
Which of the following are solutions to the given equation, where is a constant and ?
I and II only
I and III only
II and III only
I, II, and III
Choice C is correct. Subtracting the expression from both sides of the given equation yields , which can be rewritten as . Since the two terms on the right-hand side of this equation have a common factor of , it can be rewritten as , or . Since is equivalent to , the equation can be rewritten as . By the zero product property, it follows that or . Adding to both sides of the equation yields . Adding to both sides of the equation yields . Therefore, the two solutions to the given equation are and . Thus, only and , not , are solutions to the given equation.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Which of the following expressions is equivalent to ?
Choice D is correct. Expanding the given expression using the distributive property yields . Combining like terms yields
, or
, which is equivalent to
.
Choices A and B are incorrect and may result from incorrectly combining like terms. Choice C is incorrect and may result from distributing only to a, and not to 3, in the given expression.
Which of the following is equivalent to ?
Choice D is correct. The expression can be rewritten as
. Using the distributive property, the expression yields
. Combining like terms gives
.
Choices A, B, and C are incorrect and may result from errors using the distributive property on the given expression or combining like terms.
If the given function is graphed in the xy-plane, where , what is an x-intercept of the graph?
Choice A is correct. It's given that . The x-intercepts of a graph in the xy-plane are the points where . Thus, for an x-intercept of the graph of function , . Substituting for in the equation yields . Factoring the right-hand side of this equation yields . By the zero product property, and . Subtracting from both sides of the equation yields . Adding to both sides of the equation yields . Therefore, the x-intercepts of the graph of are and . Of these two x-intercepts, only is given as a choice.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The quadratic function graphed above models a particular measure of plant diversity as a function of the elevation in a region of Switzerland. According to the model, which of the following is closest to the elevation, in meters, at which plant diversity is greatest?
13,500
3,000
1,250
250
Choice C is correct. Each point on the graph represents the elevation x, in meters, and the corresponding measure of plant diversity y in a region of Switzerland. Therefore, the point on the graph with the greatest y-coordinate represents the location that has the greatest measure of plant diversity in the region. The greatest y-coordinate of any point on the graph is approximately 13,500. The x-coordinate of that point is approximately 1,250. Therefore, the closest elevation at which the plant diversity is the greatest is 1,250 meters.
Choice A is incorrect. This value is closest to the greatest y-coordinate of any point on the graph and therefore represents the greatest measure of plant diversity, not the elevation where the greatest measure of plant diversity occurs. Choice B is incorrect. At an elevation of 3,000 meters the measure of plant diversity is approximately 4,000. Because there are points on the graph with greater y-coordinates, 4,000 can’t be the greatest measure of plant diversity, and 3,000 meters isn’t the elevation at which the greatest measure of plant diversity occurs. Choice D is incorrect. At an elevation of 250 meters, the measure of plant diversity is approximately 11,000. Because there are points on the graph with greater y-coordinates, 11,000 can’t be the greatest measure of plant diversity and 250 meters isn’t the elevation at which the greatest measure of plant diversity occurs.
Square P has a side length of inches. Square Q has a perimeter that is inches greater than the perimeter of square P. The function gives the area of square Q, in square inches. Which of the following defines ?
Choice A is correct. Let represent the side length, in inches, of square P. It follows that the perimeter of square P is inches. It's given that square Q has a perimeter that is inches greater than the perimeter of square P. Thus, the perimeter of square Q is inches greater than inches, or inches. Since the perimeter of a square is times the side length of the square, each side length of Q is , or inches. Since the area of a square is calculated by multiplying the length of two sides, the area of square Q is , or square inches. It follows that function is defined by .
Choice B is incorrect. This function represents a square with side lengths inches.
Choice C is incorrect. This function represents a square with side lengths inches.
Choice D is incorrect. This function represents a square with side lengths inches.
In the given system of equations, is a constant. The graphs of the equations in the given system intersect at exactly one point, , in the xy-plane. What is the value of ?
Choice C is correct. It's given that the graphs of the equations in the given system intersect at exactly one point, , in the xy-plane. Therefore, is the only solution to the given system of equations. The given system of equations can be solved by subtracting the second equation, , from the first equation, . This yields , or . Since the given system has only one solution, this equation has only one solution. A quadratic equation in the form , where , , and are constants, has one solution if and only if the discriminant, , is equal to zero. Substituting for , for , and for in the expression yields . Setting this expression equal to zero yields , or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Substituting for in the equation yields , or . Factoring from the right-hand side of this equation yields . Dividing both sides of this equation by yields , which is equivalent to , or . Taking the square root of both sides of this equation yields . Adding to both sides of this equation yields .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The surface area of a cube is , where a is a positive constant. Which of the following gives the perimeter of one face of the cube?
Choice B is correct. A cube has 6 faces of equal area, so if the total surface area of a cube is , then the area of one face is
. Likewise, the area of one face of a cube is the square of one of its edges; therefore, if the area of one face is
, then the length of one edge of the cube is
. Since the perimeter of one face of a cube is four times the length of one edge, the perimeter is
.
Choice A is incorrect because if the perimeter of one face of the cube is , then the total surface area of the cube is
, which is not
. Choice C is incorrect because if the perimeter of one face of the cube is 4a, then the total surface area of the cube is
, which is not
. Choice D is incorrect because if the perimeter of one face of the cube is 6a, then the total surface area of the cube is
, which is not
.
What is the y-coordinate of the y-intercept of the graph shown?
The correct answer is . A y-intercept of a graph in the xy-plane is a point on the graph where . For the graph shown, at , the corresponding value of is . Therefore, the y-coordinate of the y-intercept of the graph shown is .
The function is defined by . In the xy-plane, the graph of is the result of shifting the graph of down units. Which equation defines function ?
Choice D is correct. If the graph of is the result of shifting the graph of down units in the xy-plane, the function can be defined by an equation of the form . It’s given that and the graph of is the result of shifting the graph of down units. Substituting for and for in the equation yields .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect. This equation defines a function for which the graph of is the result of shifting the graph of up, not down, units.
The table shows three values of and their corresponding values of for the equation . In the table, is a constant. What is the value of ?
Choice B is correct. It's given that the table shows three values of and their corresponding values of for the equation . It's also given that when the corresponding value of is , and is a constant. Substituting for and for in the given equation yields , or . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The equation relates the variables , , and . Which of the following correctly expresses the value of in terms of ?
Choice D is correct. Subtracting from each side of the given equation yields . Therefore, the expression correctly expresses the value of in terms of .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The graph of the exponential function is shown, where . The y-intercept of the graph is . What is the value of ?
The correct answer is . For the graph of the exponential function shown, where , it's given that the y-intercept of the graph is . The graph intersects the y-axis at the point . Therefore, the value of is .
A solution to the given system of equations is . Which of the following is a possible value of xy ?
0
6
12
36
Choice A is correct. Solutions to the given system of equations are ordered pairs that satisfy both equations in the system. Adding the left-hand and right-hand sides of the equations in the system yields
, or
. Subtracting y from both sides of this equation yields
. Taking the square root of both sides of this equation yields
and
. Therefore, there are two solutions to this system of equations, one with an x-coordinate of 6 and the other with an x-coordinate of
. Substituting 6 for x in the second equation yields
, or
; therefore, one solution is
. Similarly, substituting
for x in the second equation yields
, or
; therefore, the other solution is
. It follows then that if
is a solution to the system, then possible values of
are
and
. Only 0 is among the given choices.
Choice B is incorrect. This is the x-coordinate of one of the solutions, . Choice C is incorrect and may result from conceptual or computational errors. Choice D is incorrect. This is the square of the x-coordinate of one of the solutions,
.
According to Moore’s law, the number of transistors included on microprocessors doubles every 2 years. In 1985, a microprocessor was introduced that had 275,000 transistors. Based on this information, in which of the following years does Moore’s law estimate the number of transistors to reach 1.1 million?
1987
1989
1991
1994
Choice B is correct. Let x be the number of years after 1985. It follows that represents the number of 2-year periods that will occur within an x-year period. According to Moore’s law, every 2 years, the number of transistors included on microprocessors is estimated to double. Therefore, x years after 1985, the number of transistors will double
times. Since the number of transistors included on a microprocessor was 275,000, or .275 million, in 1985, the estimated number of transistors, in millions, included x years after 1985 can be modeled as
. The year in which the number of transistors is estimated to be 1.1 million is represented by the value of x when
. Dividing both sides of this equation by .275 yields
, which can be rewritten as
. Since the exponential equation has equal bases on each side, it follows that the exponents must also be equal:
. Multiplying both sides of the equation
by 2 yields
. Therefore, according to Moore’s law, 4 years after 1985, or in 1989, the number of transistors included on microprocessors is estimated to reach 1.1 million.
Alternate approach: According to Moore’s law, 2 years after 1985 (in 1987), the number of transistors included on a microprocessor is estimated to be , or 550,000, and 2 years after 1987 (in 1989), the number of transistors included on microprocessors is estimated to be
, or 1,100,000. Therefore, the year that Moore’s law estimates the number of transistors on microprocessors to reach 1.1 million is 1989.
Choices A, C, and D are incorrect. According to Moore’s law, the number of transistors included on microprocessors is estimated to reach 550,000 in 1987, 2.2 million in 1991, and about 6.2 million in 1994.
The function is defined by . What is the value of ?
Choice D is correct. The value of is the value of when . Substituting for in the given function yields , or , which is equivalent to . Therefore, the value of is .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the value of , not .
The function is defined by . For which value of is ?
Choice A is correct. It's given that . Substituting for in this equation yields . Subtracting from both sides of this equation yields . Taking the square root of each side of this equation yields . It follows that when the value of is or . Only is listed among the choices.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Which expression is equivalent to , where ?
Choice B is correct. An expression in the form , where and , is equivalent to . It follows that , where , is equivalent to .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
If a is a solution of the equation above and , what is the value of a ?
The correct answer is 3. The solution to the given equation can be found by factoring the quadratic expression. The factors can be determined by finding two numbers with a sum of 1 and a product of . The two numbers that meet these constraints are 4 and
. Therefore, the given equation can be rewritten as
. It follows that the solutions to the equation are
or
. Since it is given that
, a must equal 3.
Which expression is equivalent to ?
Choice A is correct. Since each term of the given expression has a factor of , it can be rewritten as , or .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The function f is defined as shown. Which of the following graphs in the xy-plane could be the graph of ?
Choice D is correct. For the quadratic function , the vertex of the graph is
, and because the
term is positive, the vertex is the minimum of the function. Choice D is the only option that meets these conditions.
Choices A, B, and C are incorrect. The vertex of each of these graphs doesn’t correspond to the minimum of the given function.
The function models the value, in dollars, of a certain bank account by the end of each year from through , where is the number of years after . Which of the following is the best interpretation of “ is approximately equal to ” in this context?
The value of the bank account is estimated to be approximately dollars greater in than in .
The value of the bank account is estimated to be approximately dollars in .
The value, in dollars, of the bank account is estimated to be approximately times greater in than in .
The value of the bank account is estimated to increase by approximately dollars every years between and .
Choice B is correct. It’s given that the function models the value, in dollars, of a certain bank account by the end of each year from through , where is the number of years after . It follows that represents the estimated value, in dollars, of the bank account years after . Since the value of is the value of when , it follows that “ is approximately equal to ” means that is approximately equal to when . In the given context, this means that the value of the bank account is estimated to be approximately dollars years after . Therefore, the best interpretation of the statement “ is approximately equal to ” in this context is the value of the bank account is estimated to be approximately dollars in .
Choice A is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
Which expression is equivalent to ?
Choice B is correct. The expression can be written as , which is equivalent to . Distributing and to yields , or . The expression is equivalent to . Distributing and to yields , or . Therefore, the expression is equivalent to , or . Combining like terms in this expression yields .
Choice A is incorrect. Equivalent expressions must be equivalent for any value of . Substituting for in this expression yields , whereas substituting for in the given expression yields .
Choice C is incorrect. Equivalent expressions must be equivalent for any value of . Substituting for in this expression yields , whereas substituting for in the given expression yields .
Choice D is incorrect. Equivalent expressions must be equivalent for any value of . Substituting for in this expression yields , whereas substituting for in the given expression yields .
The graph gives the estimated population , in thousands, of a town years since , where . Which of the following best describes the increase in the estimated population from to ?
The estimated population at is times the estimated population at .
The estimated population at is times the estimated population at .
The estimated population at is times the estimated population at .
The estimated population at is times the estimated population at .
Choice B is correct. On the graph shown, the y-axis represents estimated population, in thousands. The graph shows that when , the y-coordinate is . Therefore, the estimated population at is thousand. The graph also shows that when , the y-coordinate is . Therefore, the estimated population at is thousand. Dividing thousand by thousand yields ; therefore, thousand is times thousand. It follows that the estimated population at is times the estimated population at .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
In the xy-plane, the graph of intersects the graph of
at the points
and
. What is the value of a ?
The correct answer is 5. The intersection points of the graphs of and
can be found by solving the system consisting of these two equations. To solve the system, substitute x for y in the first equation. This gives
. Subtracting x from both sides of the equation gives
. Factoring
out of each term on the left-hand side of the equation gives
. Therefore, the possible values for x are 0 and 5. Since
, the two intersection points are
and
. Therefore,
.
The function above models the height h, in feet, of an object above ground t seconds after being launched straight up in the air. What does the number 72 represent in the function?
The initial height, in feet, of the object
The maximum height, in feet, of the object
The initial speed, in feet per second, of the object
The maximum speed, in feet per second, of the object
Choice A is correct. The variable t represents the seconds after the object is launched. Since , this means that the height, in feet, at 0 seconds, or the initial height, is 72 feet.
Choices B, C, and D are incorrect and may be the result of misinterpreting the function in context.
What is the sum of the solutions to the given equation?
The correct answer is . The given quadratic equation can be rewritten in factored form as . Based on the zero product property, it follows that or . Adding to both sides of the equation yields . Subtracting from both sides of the equation yields . Therefore, the solutions to the given equation are and . It follows that the sum of the solutions to the given equation is , or .
A model predicts that the population of Bergen was in . The model also predicts that each year for the next years, the population increased by of the previous year's population. Which equation best represents this model, where is the number of years after , for ?
Choice D is correct. It's given that a model predicts the population of Bergen in was . The model also predicts that each year for the next years, the population increased by of the previous year's population. The predicted population in one of these years can be found by multiplying the predicted population from the previous year by . Since the predicted population in was , the predicted population year later is . The predicted population years later is this value times , which is , or . The predicted population years later is this value times , or . More generally, the predicted population, , years after is represented by the equation .
Choice A is incorrect. Substituting for in this equation indicates the predicted population in was rather than .
Choice B is incorrect. Substituting for in this equation indicates the predicted population in was rather than .
Choice C is incorrect. This equation indicates the predicted population is decreasing, rather than increasing, by each year.
The function N defined above can be used to model the number of species of brachiopods at various ocean depths d, where d is in hundreds of meters. Which of the following does the model predict?
For every increase in depth by 1 meter, the number of brachiopod species decreases by 115.
For every increase in depth by 1 meter, the number of brachiopod species decreases by 10%.
For every increase in depth by 100 meters, the number of brachiopod species decreases by 115.
For every increase in depth by 100 meters, the number of brachiopod species decreases by 10%.
Choice D is correct. The function N is exponential, so it follows that changes by a fixed percentage for each increase in d by 1. Since d is measured in hundreds of meters, it also follows that the number of brachiopod species changes by a fixed percentage for each increase in ocean depth by 100 meters. Since the base of the exponent in the model is 0.90, which is less than 1, the number of brachiopod species decreases as the ocean depth increases. Specifically, the number of brachiopod species at a depth of
meters is 90% of the number of brachiopod species at a depth of d meters. This means that for each increase in ocean depth by 100 meters, the number of brachiopod species decreases by 10%.
Choices A and C are incorrect. These describe situations where the number of brachiopod species are decreasing linearly rather than exponentially. Choice B is incorrect and results from interpreting the decrease in the number of brachiopod species as 10% for every 1-meter increase in ocean depth rather than for every 100-meter increase in ocean depth.
The graph of the given equation in the xy-plane has a y-intercept of . Which of the following equivalent equations displays the value of as a constant, a coefficient, or the base?
Choice A is correct. The y-intercept of a graph in the xy-plane is the point where . Substituting for in the given equation, , yields , which is equivalent to , or . Therefore, the graph of the given equation in the xy-plane has a y-intercept of . It follows that and . Thus, the equivalent equation displays the value of as the base.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Which expression is equivalent to ?
Choice C is correct. Factoring the denominator in the second term of the given expression gives . This expression can be rewritten with common denominators by multiplying the first term by , giving . Adding these two terms yields . Using the distributive property to rewrite this expression gives . Combining the like terms in the numerator of this expression gives .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
One of the factors of is , where is a positive constant. What is the smallest possible value of ?
The correct answer is . Since each term of the given expression, , has a factor of , the expression can be rewritten as , or . Since the values and have a sum of and a product of , the expression can be factored as . Therefore, the given expression can be factored as . It follows that the factors of the given expression are , , , and . Of these factors, only and are of the form , where is a positive constant. Therefore, the possible values of are and . Thus, the smallest possible value of is .
Which ordered pair is a solution to the given system of equations?
Choice A is correct. The solution to a system of equations is the ordered pair that satisfies all equations in the system. It's given by the first equation in the system that . Substituting for into the second equation yields , or . The x-coordinate of the solution to the system of equations can be found by subtracting from both sides of the equation , which yields . Therefore, the ordered pair is a solution to the given system of equations.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The function is defined by , where and are positive constants. The graph of in the -plane passes through the points (, ) and (, ). What is the value of ?
Choice C is correct. It’s given that the function is defined by and that the graph of in the xy-plane passes through the points and . Substituting for and for in the equation yields , or . Subtracting from both sides of this equation yields . Substituting for and for in the equation yields . Subtracting from both sides of this equation yields , which can be rewritten as . Taking the square root of both sides of this equation yields and , but because it’s given that is a positive constant, must equal . Because the value of is and the value of is , the value of is , or .
Choice A is incorrect and may result from finding the value of rather than the value of .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from correctly finding the value of as , but multiplying it by the y-value in the first ordered pair rather than by the value of .
In a city, the property tax T, in dollars, is calculated using the formula above, where P is the value of the property, in dollars. Which of the following expresses the value of the property in terms of the property tax?
Choice D is correct. To express the value of the property in terms of the property tax, the given equation must be solved for P. Multiplying both sides of the equation by 100 gives . Adding 40,000 to both sides of the equation gives
. Therefore,
.
Choice A is incorrect and may result from multiplying 40,000 by 0.01, then subtracting 400 from, instead of adding 400 to, the left-hand side of the equation. Choice B is incorrect and may result from multiplying 40,000 by 0.01. Choice C is incorrect and may result from subtracting instead of adding 40,000 from the left-hand side of the equation.
The function is defined by , where and are constants. In the xy-plane, the graph of passes through the point , and . Which of the following must be true?
Choice D is correct. It's given that . Substituting for in the equation yields . Therefore, . Since can't be negative, it follows that . It's also given that the graph of passes through the point . It follows that when , . Substituting for and for in the equation yields . By the zero product property, either or . Since , it follows that . Squaring both sides of this equation yields . Adding to both sides of this equation yields . Since and is , it follows that must be true.
Choice A is incorrect. The value of is , which must be negative.
Choice B is incorrect. The value of is , which could be , but doesn't have to be.
Choice C is incorrect and may result from conceptual or calculation errors.
The table shows three values of and their corresponding values of , where and is a quadratic function. What is the y-coordinate of the y-intercept of the graph of in the xy-plane?
The correct answer is . It's given that is a quadratic function. It follows that can be defined by an equation of the form , where , , and are constants. It's also given that the table shows three values of and their corresponding values of , where . Substituting for in this equation yields . This equation represents a quadratic relationship between and , where is either the maximum or the minimum value of , which occurs when . For quadratic relationships between and , the maximum or minimum value of occurs at the value of halfway between any two values of that have the same corresponding value of . The table shows that x-values of and correspond to the same y-value, . Since is halfway between and , the maximum or minimum value of occurs at an x-value of . The table shows that when , . It follows that and . Subtracting from both sides of the equation yields . Substituting for and for in the equation yields , or . The value of can be found by substituting any x-value and its corresponding y-value for and , respectively, in this equation. Substituting for and for in this equation yields , or . Subtracting from both sides of this equation yields , or . Dividing both sides of this equation by yields . Substituting for , for , and for in the equation yields . The y-intercept of the graph of in the xy-plane is the point on the graph where . Substituting for in the equation yields , or . This is equivalent to , so the y-intercept of the graph of in the xy-plane is . Thus, the y-coordinate of the y-intercept of the graph of in the xy-plane is .
What is a solution to the given equation?
Choice B is correct. Multiplying the left- and right-hand sides of the given equation by yields . Taking the square root of the left- and right-hand sides of this equation yields or . Of these two solutions, only is given as a choice.
Choice A is incorrect. This is a solution to the equation .
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
If the given function is graphed in the xy-plane, where , what is the y-intercept of the graph?
Choice A is correct. The x-coordinate of any y-intercept of a graph is . Substituting for in the given equation yields . Since any nonzero number raised to the power is , this gives , or . The y-intercept of the graph is, therefore, the point .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Which expression is equivalent to ?
Choice D is correct. In the given expression, the first two terms, and , are like terms. Combining these like terms yields , or . It follows that the expression is equivalent to .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The given equation relates the positive numbers , , and . Which equation correctly expresses in terms of and ?
Choice D is correct. It's given that the values of , , and are positive. Therefore, dividing each side of the given equation by yields . Subtracting from each side of this equation yields . Dividing each side of this equation by yields , or .
Choice A is incorrect. This equation is equivalent to , not .
Choice B is incorrect. This equation is equivalent to , not .
Choice C is incorrect. This equation is equivalent to , not .
Which expression is equivalent to ?
Choice D is correct. The given expression is equivalent to , which can be rewritten as . Adding like terms in this expression yields , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
What is an x-coordinate of an x-intercept of the graph of in the xy-plane?
The correct answer is either , , or . The x-intercepts of a graph in the xy-plane are the points at which the graph intersects the x-axis, or when the value of is . Substituting for in the given equation yields . Dividing both sides of this equation by yields . Applying the zero product property to this equation yields three equations: , , and . Adding to both sides of the equation yields , subtracting from both sides of the equation yields , and subtracting from both sides of the equation yields . Therefore, the x-coordinates of the x-intercepts of the graph of the given equation are , , and . Note that 14, -5, and -4 are examples of ways to enter a correct answer.
The area A, in square centimeters, of a rectangular painting can be represented by the expression , where is the width, in centimeters, of the painting. Which expression represents the length, in centimeters, of the painting?
Choice C is correct. It's given that the expression represents the area, in square centimeters, of a rectangular painting, where is the width, in centimeters, of the painting. The area of a rectangle can be calculated by multiplying its length by its width. It follows that the length, in centimeters, of the painting is represented by the expression .
Choice A is incorrect. This expression represents the width, in centimeters, of the painting, not its length, in centimeters.
Choice B is incorrect. This is the difference between the length, in centimeters, and the width, in centimeters, of the painting, not its length, in centimeters.
Choice D is incorrect. This expression represents the area, in square centimeters, of the painting, not its length, in centimeters.
A quadratic function models the height, in feet, of an object above the ground in terms of the time, in seconds, after the object is launched off an elevated surface. The model indicates the object has an initial height of feet above the ground and reaches its maximum height of feet above the ground seconds after being launched. Based on the model, what is the height, in feet, of the object above the ground seconds after being launched?
Choice C is correct. It's given that a quadratic function models the height, in feet, of an object above the ground in terms of the time, in seconds, after the object is launched off an elevated surface. This quadratic function can be defined by an equation of the form , where is the height of the object seconds after it was launched, and , , and are constants such that the function reaches its maximum value, , when . Since the model indicates the object reaches its maximum height of feet above the ground seconds after being launched, reaches its maximum value, , when . Therefore, and . Substituting for and for in the function yields . Since the model indicates the object has an initial height of feet above the ground, the value of is when . Substituting for and for in the equation yields , or . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . Therefore, the model can be represented by the equation . Substituting for in this equation yields , or . Therefore, based on the model, seconds after being launched, the height of the object above the ground is feet.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The given equation models a company’s scheduled deliveries over months, where is the estimated number of scheduled deliveries months after the end of May , where . Which statement is the best interpretation of the y-intercept of the graph of this equation in the xy-plane?
At the end of May , the estimated number of scheduled deliveries was .
At the end of May , the estimated number of scheduled deliveries was .
At the end of June , the estimated number of scheduled deliveries was .
At the end of June , the estimated number of scheduled deliveries was .
Choice B is correct. The y-intercept of a graph in the xy-plane is the point where . For the given equation, the y-intercept of the graph in the xy-plane can be found by substituting for in the equation, which yields , or . Therefore, the y-intercept of the graph is . It’s given that is the estimated number of scheduled deliveries months after the end of May . Therefore, represents months after the end of May , or the end of May . Thus, the best interpretation of the y-intercept of the graph of this equation in the xy-plane is that at the end of May , the estimated number of scheduled deliveries was .
Choice A is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
The graph of a system of a linear equation and a nonlinear equation is shown. What is the solution to this system?
Choice B is correct. The solution to the system of two equations corresponds to the point where the graphs of the equations intersect in the xy-plane. The graphs of the linear equation and the nonlinear equation shown intersect at the point . Thus, the solution to this system is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The equation is true for all
, where a and d are integers. What is the value of
?
0
1
Choice C is correct. Since the expression can be factored as
, the given equation can be rewritten as
. Since
,
is also not equal to 0, so both the numerator and denominator of
can be divided by
. This gives
. Equating the coefficient of x on each side of the equation gives
. Equating the constant terms gives
. The sum is
.
Choice A is incorrect and may result from incorrectly simplifying the equation. Choices B and D are incorrect. They are the values of d and a, respectively, not .
Which expression is equivalent to ?
Choice B is correct. The given expression is equivalent to , or . Combining like terms in this expression yields .
Choice A is incorrect. This expression is equivalent to , not .
Choice C is incorrect. This expression is equivalent to , not .
Choice D is incorrect. This expression is equivalent to , not .
Which of the following expressions is equivalent to ?
Choice D is correct. The numerator of the given expression can be rewritten in terms of the denominator, , as follows:
, which is equivalent to
. So the given expression is equivalent to
. Since the given expression is defined for
, the expression can be rewritten as
.
Long division can also be used as an alternate approach. Choices A, B, and C are incorrect and may result from errors made when dividing the two polynomials or making use of structure.
A company opens an account with an initial balance of . The account earns interest, and no additional deposits or withdrawals are made. The account balance is given by an exponential function , where is the account balance, in dollars, years after the account is opened. The account balance after years is . Which equation could define ?
Choice A is correct. Since it's given that the account balance, , in dollars, after years can be modeled by an exponential function, it follows that function can be written in the form , where is the initial value of the function and is a constant related to the growth of the function. It's given that the initial balance of the account is , so it follows that the initial value of the function, or , must be . Substituting for in the equation yields . It's given that the account balance after years, or when , is . It follows that , or . Dividing each side of the equation by yields . Taking the th root of both sides of this equation yields , or is approximately equal to . Substituting for in the equation yields , so the equation could define .
Choice B is incorrect. Substituting for in this function indicates an initial balance of , rather than .
Choice C is incorrect. Substituting for in this function indicates an initial balance of , rather than . Additionally, this function indicates the account balance is decreasing, rather than increasing, over time.
Choice D is incorrect. This function indicates the account balance is decreasing, rather than increasing, over time.
Which of the following is equivalent to the sum of and
?
Choice D is correct. Adding the two expressions yields . Because the pair of terms
and
and the pair of terms
and
each contain the same variable raised to the same power, they are like terms and can be combined as
and
, respectively. The sum of the given expressions therefore simplifies to
.
Choice A is incorrect and may result from adding the coefficients and the exponents in the given expressions. Choice B is incorrect and may result from adding the exponents as well as the coefficients of the like terms in the given expressions. Choice C is incorrect and may result from multiplying, rather than adding, the coefficients of the like terms in the given expressions.
An investment account was opened with an initial value of . The value of the account doubled every years. Which equation represents the value of the account , in dollars, years after the account was opened?
Choice C is correct. It's given that represents the number of years since the account was opened. Therefore, represents the number of -year periods since the account was opened. Since the value of the account doubles during each of these -year periods, the value of the account can be found by multiplying the initial value by factors of . This is equivalent to . It's given that the initial value of the account is . Therefore, the value of the account , in dollars, years after the account was opened can be represented by .
Choice A is incorrect. This equation represents the value of an account if the value of the account halves, not doubles, every years.
Choice B is incorrect. This equation represents the value of an account if the value of the account decreases by , not doubles, every , not , years.
Choice D is incorrect. This equation represents the value of an account if the value of the account increases by a factor of , not doubles, every , not , years.
The expression above can be written in the form , where a and b are constants. What is the value of
?
The correct answer is 6632. Applying the distributive property to the expression yields . Then adding together
and
and collecting like terms results in
. This is written in the form
, where
and
. Therefore
.
The -intercept of the graph of in the xy-plane is . What is the value of ?
The correct answer is . It's given that the y-intercept of the graph of in the xy-plane is . Substituting for in the given equation yields , or . Thus, the value of is .
| x | f(x) |
| 0 | 5 |
| 1 | |
| 2 | |
| 3 |
The table above gives the values of the function f for some values of x. Which of the following equations could define f ?
Choice D is correct. Each choice has a function with coefficient 5 and base 2, so the exponents must be analyzed. When the input value of x increases, the output value of f(x) decreases, so the exponent must be negative. An exponent of –x yields the values in the table: ,
,
, and
.
Choices A and B are incorrect and may result from choosing equations that yield an increasing, rather than decreasing, output value of f(x) when the input value of x increases. Choice C is incorrect and may result from choosing an equation that doesn’t yield the values in the table.
The expression is equivalent to , where and are positive constants and . What is the value of ?
The correct answer is . The rational exponent property is , where , and are integers, and . This property can be applied to rewrite the given expression as , or . This expression can be rewritten by multiplying the constants, which gives . The multiplication exponent property is , where . This property can be applied to rewrite the expression as , or . Therefore, . It's given that is equivalent to ; therefore, and . It follows that . Finding a common denominator on the right-hand side of this equation gives , or . Note that 361/8, 45.12, and 45.13 are examples of ways to enter a correct answer.
If the given expression is rewritten in the form , where k is a constant, what is the value of k ?
The correct answer is 4. It’s given that can be rewritten as
; it follows that
. Expanding the left-hand side of this equation yields
. Subtracting
from both sides of this equation yields
. Using properties of equality,
and
. Either equation can be solved for k. Dividing both sides of
by
yields
. The equation
can be rewritten as
. It follows that
. Solving this equation for k also yields
. Therefore, the value of k is 4.
The formula above expresses the square of the speed v of a wave moving along a string in terms of tension T, mass m, and length L of the string. What is T in terms of m, v, and L ?
Choice A is correct. To write the formula as T in terms of m, v, and L means to isolate T on one side of the equation. First, multiply both sides of the equation by m, which gives , which simplifies to mv2 = LT. Next, divide both sides of the equation by L, which gives
, which simplifies to
.
Choices B, C, and D are incorrect and may be the result of incorrectly applying operations to each side of the equation.
The function f is defined by . What is the value of ?
The correct answer is . It’s given that the function is defined by . Substituting for in this equation yields , which is equivalent , or . Therefore, the value of is . Note that 11/4 or 2.75 are examples of ways to enter a correct answer.
In the given equation, and are positive constants. The sum of the solutions to the given equation is , where is a constant. What is the value of ?
The correct answer is . Let and represent the solutions to the given equation. Then, the given equation can be rewritten as , or . Since this equation is equivalent to the given equation, it follows that . Dividing both sides of this equation by yields , or . Therefore, the sum of the solutions to the given equation, , is equal to . Since it's given that the sum of the solutions to the given equation is , where is a constant, it follows that . Note that 1/16, .0625, 0.062, and 0.063 are examples of ways to enter a correct answer.
Alternate approach: The given equation can be rewritten as , where and are positive constants. Dividing both sides of this equation by yields . The solutions for a quadratic equation in the form , where , , and are constants, can be calculated using the quadratic formula, and . It follows that the sum of the solutions to a quadratic equation in the form is , which can be rewritten as , which is equivalent to , or . In the equation , , , and . Substituting for and for in yields , which can be rewritten as . Thus, the sum of the solutions to the given equation is . Since it's given that the sum of the solutions to the given equation is , where is a constant, it follows that .
What is one possible solution to the given equation?
The correct answer is or . By the definition of absolute value, if , then or . Adding to both sides of the equation yields . Adding to both sides of the equation yields . Thus, the given equation, , has two possible solutions, and . Note that 11 and -7 are examples of ways to enter a correct answer.
The graph shows the predicted value , in dollars, of a certain sport utility vehicle years after it is first purchased.
Which of the following is closest to the predicted value of the sport utility vehicle years after it is first purchased?
Choice B is correct. For the graph shown, the horizontal axis represents the number of years after a certain sport utility vehicle is first purchased, and the vertical axis represents the predicted value, in dollars, of the sport utility vehicle. According to the graph, years after the sport utility vehicle is purchased, the predicted value of the sport utility vehicle is between and . Of the given choices, only is between and . Therefore, is closest to the predicted value of the sport utility vehicle years after it is first purchased.
Choice A is incorrect. This is closest to the predicted value of the sport utility vehicle years after it is first purchased.
Choice C is incorrect. This is closest to the predicted value of the sport utility vehicle year after it is first purchased.
Choice D is incorrect. This is closest to the predicted value of the sport utility vehicle when it is first purchased.
Which expression represents the product of and ?
Choice D is correct. The product of and can be represented by the expression . Applying the distributive property to this expression yields , or . This expression is equivalent to , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
If (x, y) is a solution of the system of equations above and x > 0, what is the value of xy ?
1
2
3
9
Choice A is correct. Substituting for y in the second equation gives
. This equation can be solved as follows:
| Apply the distributive property. | |
| Subtract 2x and 6 from both sides of the equation. | |
| Combine like terms. | |
| Factor both terms on the left side of the equation by 2x. |
Thus, and
are the solutions to the system. Since
, only
needs to be considered. The value of y when
is
. Therefore, the value of xy is
.
Choices B, C, and D are incorrect and likely result from a computational or conceptual error when solving this system of equations.
Function is defined by . Function is defined by . The graph of in the xy-plane has x-intercepts at , , and , where , , and are distinct constants. What is the value of ?
Choice B is correct. It's given that . Since , it follows that . Combining like terms yields . Therefore, . The x-intercepts of a graph in the xy-plane are the points where . The x-coordinates of the x-intercepts of the graph of in the xy-plane can be found by solving the equation . Applying the zero product property to this equation yields three equations: , , and . Solving each of these equations for yields , , and , respectively. Therefore, the x-intercepts of the graph of are , , and . It follows that the values of , , and are , , and . Thus, the value of is , which is equal to .
Choice A is incorrect. This is the value of if .
Choice C is incorrect. This is the value of if .
Choice D is incorrect. This is the value of if .
If , which of the following is equivalent to the expression
?
Choice C is correct. Factoring –1 from the second, third, and fourth terms gives x2 – c2 – 2cd – d2 = x2 – (c2 + 2cd + d2). The expression c2 + 2cd + d2 is the expanded form of a perfect square: c2 + 2cd + d2 = (c + d)2. Therefore, x2 – (c2 + 2cd + d2) = x2 – (c + d)2. Since a = c + d, x2 – (c + d)2 = x2 – a2. Finally, because x2 – a2 is the difference of squares, it can be expanded as x2 – a2 = (x + a)(x – a).
Choices A and B are incorrect and may be the result of making an error in factoring the difference of squares x2 – a2. Choice D is incorrect and may be the result of incorrectly combining terms.
What is the sum of the solutions to the given equation?
Choice D is correct. Adding to each side of the given equation yields . To complete the square, adding , or , to each side of this equation yields , or . Taking the square root of each side of this equation yields . Adding to each side of this equation yields . Therefore, the solutions to the given equation are and . The sum of these solutions is , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The graphs of the equations in the given system of equations intersect at the point in the xy-plane. What is the value of ?
Choice D is correct. Since the graphs of the equations in the given system intersect at the point , the point represents a solution to the given system of equations. The first equation of the given system of equations states that . Substituting for in the second equation of the given system of equations yields , or . Therefore, the value of is .
Choice A is incorrect. This is the value of , not .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The function is defined by . What is the value of ?
Choice D is correct. The value of is the value of when . Substituting for in the given equation yields , which is equivalent to , or . Therefore, the value of is .
Choice A is incorrect. This would be the value of if function was defined by , not .
Choice B is incorrect. This would be the value of if function was defined by , not .
Choice C is incorrect. This would be the value of if function was defined by , not .
Which of the following is equivalent to the expression above?
Choice C is correct. Using the distributive property to multiply the terms in the parentheses yields , which is equivalent to
. Combining like terms results in
.
Choices A and D are incorrect and may result from conceptual errors when multiplying the terms in the given expression. Choice B is incorrect and may result from adding, instead of multiplying, and
.
The exponential function is defined by , where is a positive constant. If , what is the value of ?
The correct answer is . It's given that the exponential function is defined by , where is a positive constant, and . It follows that when , . Substituting for and for in the given equation yields . Dividing each side of this equation by yields . Taking the cube root of both sides of this equation gives . Substituting for and for in the equation yields , or . Therefore, the value of is .
What is the positive solution to the given equation?
The correct answer is . Multiplying both sides of the given equation by results in . Applying the distributive property of multiplication to the right-hand side of this equation results in . Subtracting from both sides of this equation results in . The right-hand side of this equation can be rewritten by factoring. The two values that multiply to and add to are and . It follows that the equation can be rewritten as . Setting each factor equal to yields two equations: and . Subtracting from both sides of the equation results in . Adding to both sides of the equation results in . Therefore, the positive solution to the given equation is .
An egg is thrown from a rooftop. The equation represents this situation, where is the height of the egg above the ground, in meters, seconds after it is thrown. According to the equation, what is the height, in meters, from which the egg was thrown?
The correct answer is . It's given that an egg is thrown from a rooftop and that the equation represents this situation, where is the height of the egg above the ground, in meters, seconds after it is thrown. If follows that the height, in meters, from which the egg was thrown is the value of when . Substituting for in the equation yields , or . Therefore, according to the equation, the height, in meters, from which the egg was thrown is .
Which of the following is equivalent to ?
Choice B is correct. Multiplying (1 – p) by each term of the polynomial within the second pair of parentheses gives (1 – p)1 = 1 – p; (1 – p)p = p – p2; (1 – p)p2 = p2 – p3; (1 – p)p3 = p3 – p4; (1 – p)p4 = p4 – p5; (1 – p)p5 = p5 – p6; and (1 – p)p6 = p6 – p7. Adding these seven expressions together and combining like terms gives 1 + (p – p) + (p2 – p2) + (p3 – p3) + (p4 – p4) + (p5 – p5) + (p6 – p6) – p7, which can be simplified to 1 – p7.
Choices A, C, and D are incorrect and may result from incorrectly identifying the highest power of p in the expressions or incorrectly combining like terms.
How many solutions are there to the system of equations above?
There are exactly 4 solutions.
There are exactly 2 solutions.
There is exactly 1 solution.
There are no solutions.
Choice C is correct. The second equation of the system can be rewritten as . Substituting
for y in the first equation gives
. This equation can be solved as shown below:
Substituting 1 for x in the equation gives
. Therefore,
is the only solution to the system of equations.
Choice A is incorrect. In the xy-plane, a parabola and a line can intersect at no more than two points. Since the graph of the first equation is a parabola and the graph of the second equation is a line, the system cannot have more than 2 solutions. Choice B is incorrect. There is a single ordered pair that satisfies both equations of the system. Choice D is incorrect because the ordered pair
satisfies both equations of the system.
A submersible device is used for ocean research. The function gives the depth below the surface of the ocean, in meters, of the submersible device minutes after collecting a sample, where . How many minutes after collecting the sample did it take for the submersible device to reach the surface of the ocean?
The correct answer is . It's given that the function gives the depth below the surface of the ocean, in meters, of the submersible device minutes after collecting a sample, where . It follows that when the submersible device is at the surface of the ocean, the value of is . Substituting for in the equation yields . Multiplying both sides of this equation by yields . Since a product of two factors is equal to if and only if at least one of the factors is , either or . Subtracting from both sides of the equation yields . Adding to both sides of the equation yields . Since , minutes after collecting the sample the submersible device reached the surface of the ocean.
Which of the following is equivalent to the given expression?
Choice C is correct. Distributing the negative sign to the terms in the second parentheses yields . This expression can be rewritten as
. Combining like terms results in
.
Choice A is incorrect and may result from not distributing the negative sign to the 7. Choice B is incorrect and may result from adding to
instead of subtracting
. Choice D is incorrect and may result from adding the product of
and x to the product of 3 and 7.
The function is defined by . What is the value of when is equal to ?
Choice A is correct. It's given that . Substituting for in this equation yields . Dividing each side of this equation by yields . Taking the cube root of each side of this equation yields . Therefore, when is equal to , the value of is .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
What are the solutions to the given equation?
Choice A is correct. The quadratic formula, , can be used to find the solutions to an equation in the form
. In the given equation,
,
, and
. Substituting these values into the quadratic formula gives
, or
.
Choice B is incorrect and may result from using as the quadratic formula. Choice C is incorrect and may result from using
as the quadratic formula. Choice D is incorrect and may result from using
as the quadratic formula.
Functions and are given, and in function , is a constant. If , what is the value of ?
Choice C is correct. Multiplying the given functions and yields . Applying the distributive property to the right-hand side of this equation yields . Applying the distributive property once again to the right-hand side of this equation yields , which is equivalent to . Factoring out from the second and third terms yields . Since the left-hand sides of and are equal, it follows that , or , and , or . Adding to each side of yields . Dividing each side of this equation by yields . Similarly, dividing each side of by yields . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The equation gives the estimated number of employees at a restaurant, where is the number of years since the restaurant opened. Which of the following is the best interpretation of the number in this context?
The estimated number of employees when the restaurant opened
The increase in the estimated number of employees each year
The number of years the restaurant has been open
The percent increase in the estimated number of employees each year
Choice A is correct. For an exponential function of the form , where and are constants, the initial value of the function—that is, the value of the function when —is and the value of the function increases by a factor of each time increases by . Since the function gives the estimated number of employees at a restaurant and is the number of years since the restaurant opened, the best interpretation of the number in this context is the estimated number of employees when , or when the restaurant opened.
Choice B is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
The kinetic energy, in joules, of an object with mass kilograms traveling at a speed of meters per second is given by the function , where . Which of the following is the best interpretation of in this context?
The object traveling at meters per second has a kinetic energy of joules.
The object traveling at meters per second has a kinetic energy of joules.
The object traveling at meters per second has a kinetic energy of joules.
The object traveling at meters per second has a kinetic energy of joules.
Choice A is correct. It's given that the kinetic energy, in joules, of an object with a mass of kilograms traveling at a speed of meters per second is given by the function , where . It follows that in the equation , is the value of , or the speed of the object, in meters per second, and is the kinetic energy, in joules, of the object at that speed. Therefore, the best interpretation of in this context is the object traveling at meters per second has a kinetic energy of joules.
Choice B is incorrect. The object traveling at meters per second has a kinetic energy of joules.
Choice C is incorrect. The object traveling at meters per second has a kinetic energy of joules.
Choice D is incorrect. The object traveling at meters per second has a kinetic energy of joules.
Which of the following is equivalent to the expression above?
Choice A is correct. By distributing the minus sign through the expression , the given expression can be rewritten as
, which is equivalent to
. Combining like terms gives
, or
.
Choice B is incorrect and may be the result of failing to distribute the minus sign appropriately through the second term and simplifying the expression . Choice C is incorrect and may be the result of multiplying the expressions
and
. Choice D is incorrect and may be the result of multiplying the expressions
and
.
What value of x satisfies the equation above?
3
Choice C is correct. Each fraction in the given equation can be expressed with the common denominator . Multiplying
by
yields
, and multiplying
by
yields
. Therefore, the given equation can be written as
. Multiplying each fraction by the denominator results in the equation
, or
. This equation can be solved by setting a quadratic expression equal to 0, then solving for x. Subtracting
from both sides of this equation yields
. The expression
can be factored, resulting in the equation
. By the zero product property,
or
. To solve for x in
, 1 can be added to both sides of the equation, resulting in
. Dividing both sides of this equation by 2 results in
. Solving for x in
yields
. However, this value of x would result in the second fraction of the original equation having a denominator of 0. Therefore,
is an extraneous solution. Thus, the only value of x that satisfies the given equation is
.
Choice A is incorrect and may result from solving but not realizing that this solution is extraneous because it would result in a denominator of 0 in the second fraction. Choice B is incorrect and may result from a sign error when solving
for x. Choice D is incorrect and may result from a calculation error.
Which of the following expressions is equivalent to the expression above?
Choice B is correct. Using the distributive property, the given expression can be rewritten as . Combining like terms, this expression can be rewritten as
, which is equivalent to
.
Choices A, C, and D are incorrect and may result from an error when applying the distributive property or an error when combining like terms.
Which of the following values of x satisfies the given equation?
4
32
128
Choice A is correct. Solving for x by taking the square root of both sides of the given equation yields or
. Of the choices given,
satisfies the given equation.
Choice B is incorrect and may result from a calculation error when solving for x. Choice C is incorrect and may result from dividing 64 by 2 instead of taking the square root. Choice D is incorrect and may result from multiplying 64 by 2 instead of taking the square root.
What is the positive solution to the given equation?
The correct answer is . The left-hand side of the given equation can be factored as . Therefore, the given equation, , can be written as . Applying the zero product property to this equation yields and . Subtracting from both sides of the equation yields . Dividing both sides of this equation by yields . Adding to both sides of the equation yields . Therefore, the two solutions to the given equation, , are and . It follows that is the positive solution to the given equation.
The given equation defines the function . For what value of does reach its minimum?
The correct answer is . The given equation can be rewritten in the form , where , , and are constants. When , is the value of for which reaches its minimum. The given equation can be rewritten as , which is equivalent to . This equation can be rewritten as , or , which is equivalent to . Therefore, , so the value of for which reaches its minimum is . Note that 25/4 and 6.25 are examples of ways to enter a correct answer.
The function is defined by . What value of satisfies ?
Choice C is correct. It's given that the function is defined by . It's also given that . Substituting for in the function yields and substituting for in the function yields . Therefore, and . Substituting for and for in the equation yields . Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields . By the definition of absolute value, if , then or . Dividing both sides of each of these equations by yields or , respectively. Thus, of the given choices, a value of that satisfies is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The function h is defined above, where a, b, and c are integer constants. If the zeros of the function are 6, and 7, what is the value of c ?
The correct answer is 210. Since , 6, and 7 are zeros of the function, the function can be rewritten as
. Expanding the function yields
. Thus,
,
, and
. Therefore, the value of c is 210.
The graph of a system of an absolute value function and a linear function is shown. What is the solution to this system of two equations?
Choice C is correct. The solution to the system of two equations corresponds to the point where the graphs of the equations intersect. The graphs of the linear function and the absolute value function shown intersect at the point . Thus, the solution to the system is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the y-intercept of the graph of the linear function.
Choice D is incorrect. This is the vertex of the graph of the absolute value function.
The function is defined by . For what value of does ?
Choice C is correct. It's given that . Substituting for in this equation yields . Dividing both sides of this equation by yields . This can be rewritten as . Squaring both sides of this equation yields . Therefore, the value of for which is .
Choice A is incorrect. If , , not .
Choice B is incorrect. If , , not .
Choice D is incorrect. If , , which is equivalent to , not .
Which of the following is equivalent to , where
?
Choice D is correct. The given expression can also be written as . The trinomial
can be rewritten in factored form as
. Thus, the entire expression can be rewritten as
. Simplifying the exponents yields
.
Choices A, B, and C are incorrect and may result from errors made when simplifying the exponents in the expression .
The function is defined by the given equation. For what value of does reach its minimum?
The correct answer is . The value of for which reaches its minimum can be found by rewriting the given equation in the form , where reaches its minimum, , when the value of is . The given equation, , can be rewritten as . By completing the square, this equation can be rewritten as , which is equivalent to , or . Therefore, reaches its minimum when the value of is . Note that -13/2 and -6.5 are examples of ways to enter a correct answer.
Alternate approach: The graph of in the xy-plane is a parabola. The value of for the vertex of a parabola is the x-value of the midpoint between the two x-intercepts of the parabola. Since it's given that , it follows that the two x-intercepts of the graph of in the xy-plane occur when and , or at the points and . The midpoint between two points, and , is . Therefore, the midpoint between and is , or . It follows that reaches its minimum when the value of is . Note that -13/2 and -6.5 are examples of ways to enter a correct answer.
The graph of is shown in the xy-plane. The value of is an integer. What is the value of ?
The correct answer is . The value of is the value of on the graph of in the xy-plane that corresponds with . It's given that the value of is an integer. For the graph of shown, when , the corresponding integer value of is . Therefore, the value of is .
In the given equation, is a constant. For which of the following values of will the equation have more than one real solution?
Choice A is correct. A quadratic equation of the form , where , , and are constants, has either no real solutions, exactly one real solution, or exactly two real solutions. That is, for the given equation to have more than one real solution, it must have exactly two real solutions. When the value of the discriminant, or , is greater than 0, the given equation has exactly two real solutions. In the given equation, , and . Therefore, the given equation has exactly two real solutions when , or . Adding to both sides of this inequality yields . Taking the square root of both sides of yields two possible inequalities: or . Of the choices, only choice A satisfies or .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The function is defined by the given equation. If , which of the following equations defines the function ?
Choice B is correct. It’s given that and . Substituting for in gives . Rewriting this equation using properties of exponents gives , which is equivalent to . Multiplying and in this equation gives . Since , .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
If a and c are positive numbers, which of the following is equivalent to ?
Choice C is correct. Using the property that for positive numbers x and y, with x = (a + c)3 and y = a + c, it follows that
. By rewriting (a + c)4 as ((a + c)2)2, it is possible to simplify the square root expression as follows:
.
Choice A is incorrect and may be the result of . Choice B is incorrect and may be the result of incorrectly rewriting (a + c)2 as a2 + c2. Choice D is incorrect and may be the result of incorrectly applying properties of exponents.
The functions and are defined by the given equations, where . Which of the following equations displays, as a constant or coefficient, the maximum value of the function it defines, where ?
I only
II only
I and II
Neither I nor II
Choice B is correct. For the function , since the base of the exponent, , is greater than , the value of increases as increases. Therefore, the value of and the value of also increase as increases. Since is therefore an increasing function where , the function has no maximum value. For the function , since the base of the exponent, , is less than , the value of decreases as increases. Therefore, the value of also decreases as increases. It follows that the maximum value of for occurs when . Substituting for in the function yields , which is equivalent to , or . Therefore, the maximum value of for is , which appears as a coefficient in equation II. So, of the two equations given, only II displays, as a constant or coefficient, the maximum value of the function it defines, where .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
A rectangle has an area of square inches. The length of the rectangle is inches less than times the width of the rectangle. What is the width of the rectangle, in inches?
The correct answer is . Let represent the width, in inches, of the rectangle. It's given that the length of the rectangle is inches less than times its width, or inches. The area of a rectangle is equal to its width multiplied by its length. Multiplying the width, inches, by the length, inches, yields square inches. It’s given that the rectangle has an area of square inches, so it follows that , or . Subtracting from both sides of this equation yields . Factoring the left-hand side of this equation yields . Applying the zero product property to this equation yields two solutions: and . Since is the rectangle’s width, in inches, which must be positive, the value of is . Therefore, the width of the rectangle, in inches, is .
The function is defined by the given equation. For what value of does reach its minimum?
The correct answer is . For a quadratic function defined by an equation of the form , where , , and are constants and , the function reaches its minimum when . In the given function, , , and . Therefore, the value of for which reaches its minimum is .
The product of a positive number and the number that is more than is . What is the value of ?
Choice B is correct. The number that's more than can be represented by the expression . It's given that the product of and is , so it follows that , or . Subtracting from each side of this equation yields . Factoring the left-hand side of this equation yields . Applying the zero product property to this equation yields two solutions: and . Since is a positive number, the value of is .
Choice A is incorrect. If , the product of and the number that's more than would be , or , not .
Choice C is incorrect. This is the value of the number that's more than , not the value of .
Choice D is incorrect. If , the product of and the number that's more than would be , or , not .
The solution to the given system of equations is . What is the value of ?
The correct answer is . In the given system of equations, the second equation is . Subtracting from both sides of this equation yields . In the given system of equations, the first equation is . Substituting for in this equation yields , or . Therefore, the solution to the given system of equations is . Thus, the value of is .
Which expression is equivalent to ?
Choice A is correct. Since each term of the given expression has a common factor of , it may be rewritten as . Therefore, the expression is equivalent to .
Choice B is incorrect. This expression is equivalent to , not .
Choice C is incorrect. This expression is equivalent to , not .
Choice D is incorrect. This expression is equivalent to , not .
The total revenue from sales of a product can be calculated using the formula , where T is the total revenue, P is the price of the product, and Q is the quantity of the product sold. Which of the following equations gives the quantity of product sold in terms of P and T ?
Choice B is correct. Solving the given equation for Q gives the quantity of the product sold in terms of P and T. Dividing both sides of the given equation by P yields , or
. Therefore,
gives the quantity of product sold in terms of P and T.
Choice A is incorrect and may result from an error when dividing both sides of the given equation by P. Choice C is incorrect and may result from multiplying, rather than dividing, both sides of the given equation by P. Choice D is incorrect and may result from subtracting P from both sides of the equation rather than dividing both sides by P.
The function is defined by . What is the value of ?
Choice C is correct. The value of is the value of when . Substituting for in the given function yields , or , which is equivalent to . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the value of when , rather than .
Choice D is incorrect and may result from conceptual or calculation errors.
The function is defined by the given equation. If , which of the following equations defines function ?
Choice A is correct. It's given that and . Substituting for in yields . Substituting for in this equation yields . Using the properties of exponents, the function can be rewritten as , which is equivalent to , or . Therefore, of the given choices, an equation that defines function is .
Choice B is incorrect. This equation defines function if , not .
Choice C is incorrect. This equation defines function if , not .
Choice D is incorrect and may result from conceptual or calculation errors.
The y-intercept of the graph shown is . What is the value of ?
The correct answer is . The y-intercept of a graph in the xy-plane is the point on the graph where . The y-intercept of the graph shown is . Therefore, the value of is .
The graph of the polynomial function , where , is shown. The y-intercept of the graph is . What is the value of ?
The correct answer is . The y-intercept of the graph of a function in the xy-plane is the point where the graph crosses the y-axis. The graph of the polynomial function shown crosses the y-axis at the point . It's given that the y-intercept of the graph is . Thus, the value of is .
In the given system of equations, is a positive constant. The system has exactly one distinct real solution. What is the value of ?
The correct answer is . According to the first equation in the given system, the value of is . Substituting for in the second equation in the given system yields . Adding to both sides of this equation yields . If the given system has exactly one distinct real solution, it follows that has exactly one distinct real solution. A quadratic equation in the form , where , , and are constants, has exactly one distinct real solution if and only if the discriminant, , is equal to . The equation is in this form, where , , and . Therefore, the discriminant of the equation is , or . Setting the discriminant equal to to solve for yields . Adding to both sides of this equation yields . Dividing both sides of this equation by yields , or . Therefore, if the given system of equations has exactly one distinct real solution, the value of is . Note that 29/2 and 14.5 are examples of ways to enter a correct answer.
The function f is defined above. Which of the following is NOT an x-intercept of the graph of the function in the xy-plane?
Choice B is correct. The graph of the function f in the xy-plane has x-intercepts at the points , where
. Substituting 0 for
in the given equation yields
. By the zero product property, if
, then
,
, or
. Solving each of these linear equations for x, it follows that
,
, and
, respectively. This means that the graph of the function f in the xy-plane has three x-intercepts:
,
, and
. Therefore,
isn’t an x-intercept of the graph of the function f.
Alternate approach: Substitution may be used. Since by definition an x-intercept of any graph is a point in the form where k is a constant, and since all points in the options are in this form, it need only be checked whether the points in the options lie on the graph of the function f. Substituting
for x and 0 for
in the given equation yields
, or
. Therefore, the point
doesn’t lie on the graph of the function f and can’t be an x-intercept of the graph.
Choices A, C, and D are incorrect because each of these points is an x-intercept of the graph of the function f in the xy-plane. By definition, an x-intercept is a point on the graph of the form , where k is a constant. Substituting
for x and 0 for
in the given equation yields
, or
. Since this is a true statement, the point
lies on the graph of the function f and is an x-intercept of the graph. Performing similar substitution using the points
and
also yields the true statement
, illustrating that these points also lie on the graph of the function f and are x-intercepts of the graph.
Which expression is equivalent to ?
Choice B is correct. The given expression can be rewritten as . Adding and in this expression yields .
Choice A is incorrect. This expression is equivalent to .
Choice C is incorrect. This expression is equivalent to .
Choice D is incorrect. This expression is equivalent to .
The graph of the exponential function is shown, where . The y-intercept of the graph is . What is the value of ?
The correct answer is . It's given that the y-intercept of the graph shown is . The graph passes through the point . Therefore, the value of is .
The quadratic function models the depth, in meters, below the surface of the water of a seal minutes after the seal entered the water during a dive. The function estimates that the seal reached its maximum depth of meters minutes after it entered the water and then reached the surface of the water minutes after it entered the water. Based on the function, what was the estimated depth, to the nearest meter, of the seal minutes after it entered the water?
The correct answer is . The quadratic function gives the estimated depth of the seal, , in meters, minutes after the seal enters the water. It's given that function estimates that the seal reached its maximum depth of meters minutes after it entered the water. Therefore, function can be expressed in vertex form as , where is a constant. Since it's also given that the seal reached the surface of the water after minutes, . Substituting for and for in yields , or . Dividing both sides of this equation by gives . Substituting for in gives . Substituting for in gives , which is equivalent to , or . Therefore, the estimated depth, to the nearest meter, of the seal minutes after it entered the water was meters.
At the time that an article was first featured on the home page of a news website, there were comments on the article. An exponential model estimates that at the end of each hour after the article was first featured on the home page, the number of comments on the article had increased by of the number of comments on the article at the end of the previous hour. Which of the following equations best represents this model, where is the estimated number of comments on the article hours after the article was first featured on the home page and ?
Choice D is correct. It's given that an exponential model estimates that the number of comments on an article increased by a fixed percentage at the end of each hour. Therefore, the model can be represented by an exponential equation of the form , where is the estimated number of comments on the article hours after the article was first featured on the home page and and are constants. It's also given that when the article was first featured on the home page of the news website, there were comments on the article. This means that when , . Substituting for and for in the equation yields , or . It's also given that the number of comments on the article at the end of an hour had increased by of the number of comments on the article at the end of the previous hour. Multiplying the percent increase by the number of comments on the article at the end of the previous hour yields the number of estimated additional comments the article has on its home page: , or comments. Thus, the estimated number of comments for the following hour is the sum of the comments from the end of the previous hour and the number of additional comments, which is , or . This means that when , . Substituting for , for , and for in the equation yields , or . Dividing both sides of this equation by yields . Substituting for and for in the equation yields . Thus, the equation that best represents this model is .
Choice A is incorrect. This model represents a situation where the number of comments at the end of each hour increased by of the number of comments at the end of the previous hour, rather than .
Choice B is incorrect. This model represents a situation where the number of comments at the end of each hour increased by of the number of comments at the end of the previous hour, rather than .
Choice C is incorrect. This model represents a situation where the number of comments at the end of each hour was times the number of comments at the end of the previous hour, rather than increasing by of the number of comments at the end of the previous hour.
Which of the following expressions is equivalent to ?
Choice B is correct. Applying the distributive property to the given expression yields , or
. Adding the like terms
and 2 results in the expression
.
Choice A is incorrect and may result from multiplying by 2 without multiplying
by 2 when applying the distributive property. Choices C and D are incorrect and may result from computational or conceptual errors.
The graph of the rational function is shown, where and . Which of the following is the graph of , where ?
Choice D is correct. It's given that the graph of the rational function is shown, where and . The graph shown passes through the point . It follows that when the value of is , the value of is . When the value of is , the value of is , or . Therefore, the graph of passes through the point . Of the given choices, choice D is the only graph that passes through the point and is therefore the graph of .
Choice A is incorrect. This is the graph of , rather than .
Choice B is incorrect. This is the graph of , rather than .
Choice C is incorrect and may result from conceptual or calculation errors.
If , what is the value of ?
Choice C is correct. Multiplying both sides of the given equation by yields . Therefore, the value of is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The given function models the population of Lowell years after a census. Which of the following functions best models the population of Lowell months after the census?
Choice D is correct. It’s given that the function models the population of Lowell years after a census. Since there are months in a year, months after the census is equivalent to years after the census. Substituting for in the equation yields . Therefore, the function that best models the population of Lowell months after the census is .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The given equation relates the positive numbers , , and . Which equation correctly expresses in terms of and ?
Choice D is correct. Multiplying both sides of the given equation by yields . Distributing on the left-hand side of this equation yields , or . Therefore, the equation correctly expresses in terms of and .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
If is a factor of the expression , where and are constants and , what is the value of ?
Choice D is correct. If is a factor of the expression , then the expression can be written as , where and are constants. This expression can be rewritten as , or . Since this expression is equivalent to , it follows that , , and . Dividing each side of the equation by yields . Substituting for in the equation yields . Adding to each side of this equation yields . Substituting for in the equation yields . Since is positive, dividing each side of this equation by yields . Multiplying each side of this equation by yields .
Alternate approach: The expression can be written as , which is a difference of two squares. It follows that is equivalent to . It’s given that is a factor of , so the factor is equal to . Adding to both sides of the equation yields . Since is positive, dividing both sides of this equation by yields . Squaring both sides of this equation yields . Multiplying both sides of this equation by yields .
Choice A is incorrect. This value of gives the expression , or . This expression doesn't have as a factor.
Choice B is incorrect. This value of gives the expression , or . This expression doesn't have as a factor.
Choice C is incorrect. This value of gives the expression , or . This expression doesn't have as a factor.
The population of a town is currently 50,000, and the population is estimated to increase each year by 3% from the previous year. Which of the following equations can be used to estimate the number of years, t, it will take for the population of the town to reach 60,000 ?
Choice D is correct. Stating that the population will increase each year by 3% from the previous year is equivalent to saying that the population each year will be 103% of the population the year before. Since the initial population is 50,000, the population after t years is given by 50,000(1.03)t. It follows that the equation 60,000 = 50,000(1.03)t can be used to estimate the number of years it will take for the population to reach 60,000.
Choice A is incorrect. This equation models how long it will take the population to decrease from 60,000 to 50,000, which is impossible given the growth factor. Choice B is incorrect and may result from misinterpreting a 3% growth as growth by a factor of 3. Additionally, this equation attempts to model how long it will take the population to decrease from 60,000 to 50,000. Choice C is incorrect and may result from misunderstanding how to model percent growth by multiplying the initial amount by a factor greater than 1.
For the exponential function , the value of is , where is a constant. Which of the following equivalent forms of the function shows the value of as the coefficient or the base?
Choice C is correct. For the form of the function in choice C, , the value of can be found as , which is equivalent to , or . Therefore, , which is shown in as the coefficient.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Which expression is equivalent to ?
Choice D is correct. Expressions in the form follow the difference of two squares pattern and can be factored as . In the given expression, , the expression follows the difference of two squares pattern. It follows that the expression can be rewritten as . Therefore, the expression is equivalent to .
Choice A is incorrect. This expression is equivalent to , not .
Choice B is incorrect. This expression is equivalent to , not .
Choice C is incorrect. This expression is equivalent to , not .
An object is kicked from a platform. The equation represents this situation, where is the height of the object above the ground, in meters, seconds after it is kicked. Which number represents the height, in meters, from which the object was kicked?
Choice D is correct. It’s given that the equation represents this situation, where is the height, in meters, of the object seconds after it is kicked. It follows that the height, in meters, from which the object was kicked is the value of when . Substituting for in the equation yields , or . Therefore, the object was kicked from a height of meters.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The given equation relates the positive numbers , , and . Which equation correctly expresses in terms of and ?
Choice B is correct. Adding to each side of the given equation yields . Therefore, the equation correctly expresses in terms of and .
Choice A is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
The given expression is equivalent to , where is a constant. What is the value of ?
The correct answer is . The given expression can be rewritten as . By applying the distributive property, this expression can be rewritten as , which is equivalent to . Adding like terms in this expression yields . Since it's given that is equivalent to , it follows that is equivalent to . Therefore, the coefficients of in these two expressions must be equivalent, and the value of must be .
Which expression is equivalent to ?
Choice D is correct. Combining like terms in the given expression yields , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The quadratic function above models the height above the ground h, in feet, of a projectile x seconds after it had been launched vertically. If is graphed in the xy-plane, which of the following represents the real-life meaning of the positive x-intercept of the graph?
The initial height of the projectile
The maximum height of the projectile
The time at which the projectile reaches its maximum height
The time at which the projectile hits the ground
Choice D is correct. The positive x-intercept of the graph of is a point
for which
. Since
models the height above the ground, in feet, of the projectile, a y-value of 0 must correspond to the height of the projectile when it is 0 feet above ground or, in other words, when the projectile is on the ground. Since x represents the time since the projectile was launched, it follows that the positive x-intercept,
, represents the time at which the projectile hits the ground.
Choice A is incorrect and may result from misidentifying the y-intercept as a positive x-intercept. Choice B is incorrect and may result from misidentifying the y-value of the vertex of the graph of the function as an x-intercept. Choice C is incorrect and may result from misidentifying the x-value of the vertex of the graph of the function as an x-intercept.
The graph of a system of an absolute value function and a linear function is shown. What is the solution to this system of two equations?
Choice C is correct. The solution to a system of two equations corresponds to the point where the graphs of the equations intersect. The graphs of the linear function and the absolute value function shown intersect at a point with an x-coordinate between and and a y-coordinate between and . Of the given choices, only has an x-coordinate between and and a y-coordinate between and .
Choice A is incorrect. This is the y-intercept of the graph of the linear function.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. This is the vertex of the graph of the absolute value function.
A solution to the given system of equations is , where . What is the value of ?
The correct answer is . It’s given that and . Since , substituting for in the second equation of the given system yields . Subtracting from both sides of this equation yields . This equation can be rewritten as . By the zero product property, or . Adding to both sides of the equation yields . Subtracting from both sides of the equation yields . Therefore, solutions to the given system of equations occur when and when . It’s given that a solution to the given system of equations is , where . Since is greater than , it follows that the value of is .
When the equations above are graphed in the xy-plane, what are the coordinates (x, y) of the points of intersection of the two graphs?
and
and
and
and
Choice A is correct. The two equations form a system of equations, and the solutions to the system correspond to the points of intersection of the graphs. The solutions to the system can be found by substitution. Since the second equation gives y = 3, substituting 3 for y in the first equation gives 3 = x2 – 1. Adding 1 to both sides of the equation gives 4 = x2. Solving by taking the square root of both sides of the equation gives x = ±2. Since y = 3 for all values of x for the second equation, the solutions are (2, 3) and (–2, 3). Therefore, the points of intersection of the two graphs are (2, 3) and (–2, 3).
Choices B, C, and D are incorrect and may be the result of calculation errors.
The function is defined by . What is the value of ?
Choice D is correct. It's given that the function is defined by . Substituting for in yields , which is equivalent to , or . Therefore, the value of is .
Choice A is incorrect. This is the value of if .
Choice B is incorrect. This is the value of if .
Choice C is incorrect. This is the value of if .
Which expression is equivalent to ?
Choice B is correct. Since is a common factor of each term in the given expression, the expression can be rewritten as .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This expression is equivalent to .
Choice D is incorrect. This expression is equivalent to .
If , what is the positive value of ?
The correct answer is . The given absolute value equation can be rewritten as two linear equations: and , or . Adding to both sides of the equation results in . Dividing both sides of this equation by results in . Adding to both sides of the equation results in . Dividing both sides of this equation by results in . Therefore, the two values of are , or , and , or . Thus, the positive value of is .
Alternate approach: The given equation can be rewritten as , which is equivalent to . Dividing both sides of this equation by yields . This equation can be rewritten as two linear equations: and , or . Therefore, the positive value of is .
For the exponential function f, the table above shows several values of x and their corresponding values of , where a is a constant greater than 1. If k is a constant and
, what is the value of k ?
The correct answer is 8. The values of for the exponential function f shown in the table increase by a factor of
for each increase of 1 in x. This relationship can be represented by the equation
, where b is a constant. It’s given that when
,
. Substituting 2 for x and
for
into
yields
. Since
, it follows that
. Thus, an equation that defines the function f is
. It follows that the value of k such that
can be found by solving the equation
, which yields
.
Which expression is equivalent to ?
Choice D is correct. The given expression follows the difference of two squares pattern, , which factors as . Therefore, the expression can be written as , or , which factors as .
Choice A is incorrect. This expression is equivalent to .
Choice B is incorrect. This expression is equivalent to .
Choice C is incorrect. This expression is equivalent to .
Which table gives three values of and their corresponding values of for the given function ?
Choice B is correct. It′s given that . Each table gives , , and as the three given values of . Substituting for in the equation yields , or . Substituting for in the equation yields , or . Finally, substituting for in the equation yields , or . Therefore, is when is , is when is , and is when is . Choice B is a table with these values of and their corresponding values of .
Choice A is incorrect. This is a table of values for the function , not .
Choice C is incorrect. This is a table of values for the function , not .
Choice D is incorrect and may result from conceptual or calculation errors.
Bacteria are growing in a liquid growth medium. There were cells per milliliter during an initial observation. The number of cells per milliliter doubles every hours. How many cells per milliliter will there be hours after the initial observation?
Choice D is correct. Let represent the number of cells per milliliter hours after the initial observation. Since the number of cells per milliliter doubles every hours, the relationship between and can be represented by an exponential equation of the form , where is the number of cells per milliliter during the initial observation and the number of cells per milliliter increases by a factor of every hours. It’s given that there were cells per milliliter during the initial observation. Therefore, . It’s also given that the number of cells per milliliter doubles, or increases by a factor of , every hours. Therefore, and . Substituting for , for , and for in the equation yields . The number of cells per milliliter there will be hours after the initial observation is the value of in this equation when . Substituting for in the equation yields , or . This is equivalent to , or . Therefore, hours after the initial observation, there will be cells per milliliter.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Which of the following is equivalent to the given expression?
Choice A is correct. Since 5 is a factor of both terms, and 15, the given expression can be factored and rewritten as
.
Choice B is incorrect and may result from subtracting 5 from the constant when factoring 5 from the given expression. Choice C is incorrect and may result from factoring 5 from only the first term, not both terms, of the given expression. Choice D is incorrect and may result from adding 5 to the constant when factoring 5 from the given expression.
The rational function is defined by an equation in the form , where and are constants. The partial graph of is shown. If , which equation could define function ?
Choice C is correct. It's given that and that the graph shown is a partial graph of . Substituting for in the equation yields . The graph passes through the point . Substituting for and for in the equation yields . Multiplying each side of this equation by yields , or . The graph also passes through the point . Substituting for and for in the equation yields . Multiplying each side of this equation by yields , or . Substituting for in this equation yields . Adding to each side of this equation yields . Subtracting from each side of this equation yields . Dividing each side of this equation by yields . Substituting for in the equation yields , or . Substituting for and for in the equation yields . It's given that . Substituting for in the equation yields , which is equivalent to . It follows that .
Choice A is incorrect. This could define function if .
Choice B is incorrect. This could define function if .
Choice D is incorrect. This could define function if .
The function is defined by . What is the value of when ?
Choice A is correct. It's given that . Substituting for in this function yields . Therefore, when , the value of is .
Choice B is incorrect. This is the value of the reciprocal of when .
Choice C is incorrect. This is the value of when .
Choice D is incorrect. This is the value of when .
The area of a triangle is equal to square centimeters. The length of the base of the triangle is centimeters, and the height of the triangle is centimeters. What is the value of ?
The correct answer is . The area of a triangle is equal to one half of the product of the length of the base of the triangle and the height of the triangle. It's given that the length of the base of the triangle is centimeters and the height of the triangle is centimeters; therefore, its area is square centimeters. It's also given that the area of the triangle is equal to square centimeters. Therefore, it follows that . This equation can be rewritten as , or . Subtracting from both sides of this equation yields . Adding to both sides of this equation yields . Therefore, the value of is .
The function is defined by the given equation. For what value of does reach its minimum?
Choice D is correct. It's given that , which can be rewritten as . Since the coefficient of the -term is positive, the graph of in the xy-plane opens upward and reaches its minimum value at its vertex. The x-coordinate of the vertex is the value of such that reaches its minimum. For an equation in the form , where , , and are constants, the x-coordinate of the vertex is . For the equation , , , and . It follows that the x-coordinate of the vertex is , or . Therefore, reaches its minimum when the value of is .
Alternate approach: The value of for the vertex of a parabola is the x-value of the midpoint between the two x-intercepts of the parabola. Since it’s given that , it follows that the two x-intercepts of the graph of in the xy-plane occur when and , or at the points and . The midpoint between two points, and , is . Therefore, the midpoint between and is , or . It follows that reaches its minimum when the value of is .
Choice A is incorrect. This is the y-coordinate of the y-intercept of the graph of in the xy-plane.
Choice B is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of in the xy-plane.
Choice C is incorrect and may result from conceptual or calculation errors.
Which expression is equivalent to , where and are positive?
Choice B is correct. For positive values of and , , , and . Therefore, the given expression, , can be rewritten as . This expression is equivalent to , which can be rewritten as , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The given equation relates the variables , , and , where , , and . Which expression is equivalent to ?
Choice D is correct. Adding to each side of the given equation yields . The fractions on the right side of this equation have a common denominator of ; therefore, the equation can be written as , or , which is equivalent to . Dividing each side of this equation by yields . Since , , , and are all positive quantities, taking the reciprocal of each side of the equation yields an equivalent equation: . Multiplying each side of this equation by yields .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is equivalent to rather than .
The function gives the predicted body mass , in , of a certain animal days after it was born in a wildlife reserve, where . Which of the following is the best interpretation of the statement “ is approximately equal to ” in this context?
The predicted body mass of the animal was approximately days after it was born.
The predicted body mass of the animal was approximately days after it was born.
The predicted body mass of the animal was approximately days after it was born.
The predicted body mass of the animal was approximately days after it was born.
Choice B is correct. In the statement " is approximately equal to ," the input of the function, , is the value of , the elapsed time, in days, since the animal was born. The approximate value of the function, , is the predicted body mass, in kilograms, of the animal after that time has elapsed. Therefore, the predicted body mass of the animal was approximately days after it was born.
Choice A is incorrect. This would be the best interpretation of the statement " is approximately equal to ."
Choice C is incorrect. The number is the number of weeks, not the number of days, after the animal was born.
Choice D is incorrect. This would be the best interpretation of the statement " is approximately equal to ."
In the given equation, and are unique positive constants. When the equation is graphed in the xy-plane, how many distinct x-intercepts does the graph have?
Choice B is correct. An x-intercept of a graph in the xy-plane is a point on the graph where the value of is . Substituting for in the given equation yields . By the zero product property, the solutions to this equation are , , , and . However, and are identical. It's given that and are unique positive constants. It follows that the equation has unique solutions: , , and . Thus, the graph of the given equation has distinct x-intercepts.
Choice A is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from conceptual errors.
Choice D is incorrect and may result from conceptual errors.
The functions and are defined by the given equations, where . Which of the following equations displays, as a constant or coefficient, the minimum value of the function it defines, where ?
I only
II only
I and II
Neither I nor II
Choice D is correct. A function defined by an equation in the form , where , , and are positive constants and , has a minimum value of . It's given that function is defined by , which is equivalent to . Substituting for in this equation yields , or . Therefore, the minimum value of is , so doesn't display its minimum value as a constant or coefficient. It's also given that function is defined by . Substituting for in this equation yields , or . Therefore, the minimum value of is , so doesn't display its minimum value as a constant or coefficient. Therefore, neither I nor II displays, as a constant or coefficient, the minimum value of the function it defines, where .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The area , in square centimeters, of a rectangular cutting board can be represented by the expression , where is the width, in centimeters, of the cutting board. Which expression represents the length, in centimeters, of the cutting board?
Choice D is correct. It's given that the expression represents the area, in square centimeters, of a rectangular cutting board, where is the width, in centimeters, of the cutting board. The area of a rectangle can be calculated by multiplying its length by its width. It follows that the length, in centimeters, of the cutting board is represented by the expression .
Choice A is incorrect. This expression represents the area, in square centimeters, of the cutting board, not its length, in centimeters.
Choice B is incorrect. This expression represents the width, in centimeters, of the cutting board, not its length.
Choice C is incorrect. This is the difference between the length, in centimeters, and the width, in centimeters, of the cutting board, not its length, in centimeters.
Which expression is equivalent to ?
Choice B is correct. The expression can be rewritten as , which is equivalent to .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
For the function , the value of decreases by for every increase in the value of by . If , which equation defines ?
Choice C is correct. Since the value of decreases by a fixed percentage, , for every increase in the value of by , the function is a decreasing exponential function. A decreasing exponential function can be written in the form , where is the value of and the value of decreases by for every increase in the value of by . If , then . Since the value of decreases by for every increase in the value of by , . Substituting for and for in the equation yields , which is equivalent to , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect. For this function, the value of increases, rather than decreases, by for every increase in the value of by .
In the xy-plane, the graph of the equation intersects the line at exactly one point. What is the value of ?
Choice C is correct. In the xy-plane, the graph of the line is a horizontal line that crosses the y-axis at and the graph of the quadratic equation is a parabola. A parabola can intersect a horizontal line at exactly one point only at its vertex. Therefore, the value of should be equal to the y-coordinate of the vertex of the graph of the given equation. For a quadratic equation in vertex form, , the vertex of its graph in the xy-plane is . The given quadratic equation, , can be rewritten as , or . Thus, the value of is equal to .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Which of the following is equivalent to the expression above?
Choice B is correct. Expanding all terms yields (x – 11y)(2x – 3y) – 12y(–2x + 3y), which is equivalent to 2x2 – 22xy – 3xy + 33y2 + 24xy – 36y2. Combining like terms gives 2x2 – xy – 3y2.
Choice A is incorrect and may be the result of using the sums of the coefficients of the existing x and y terms as the coefficients of the x and y terms in the new expressions. Choice C is incorrect and may be the result of incorrectly combining like terms. Choice D is incorrect and may be the result of using the incorrect sign in front of the 12y term.
The given equation relates the positive numbers , , and . Which equation correctly expresses in terms of and ?
Choice A is correct. It's given that the equation relates the positive numbers , , and . Dividing each side of the given equation by yields , or . Thus, the equation correctly expresses in terms of and .
Choice B is incorrect. This equation can be rewritten as .
Choice C is incorrect. This equation can be rewritten as .
Choice D is incorrect. This equation can be rewritten as .
Two variables, and , are related such that for each increase of in the value of , the value of increases by a factor of . When , . Which equation represents this relationship?
Choice D is correct. Since the value of increases by a constant factor, , for each increase of in the value of , the relationship between and is exponential. An exponential relationship between and can be represented by an equation of the form , where is the value of when and increases by a factor of for each increase of in the value of . Since when , . Since increases by a factor of for each increase of in the value of , . Substituting for and for in the equation yields . Thus, the equation represents the relationship between and .
Choice A is incorrect and may result from conceptual errors.
Choice B is incorrect. This equation represents a relationship where for each increase of in the value of , the value of increases by a factor of , not , and when , is equal to , not .
Choice C is incorrect and may result from conceptual errors.
The graph of the quadratic function is shown. What is the vertex of the graph?
Choice C is correct. The vertex of the graph of a quadratic function in the xy-plane is the point at which the graph is either at its minimum or maximum y-value. In the graph shown, the minimum y-value occurs at the point .
Choice A is incorrect. The graph shown doesn't pass through the point .
Choice B is incorrect. The graph shown doesn't pass through the point .
Choice D is incorrect. The graph shown doesn't pass through the point .
Which expression is equivalent to ?
Choice C is correct. The given expression shows subtraction of two like terms. The two terms can be subtracted as follows: , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the result of adding, not subtracting, the two like terms.
Choice D is incorrect and may result from conceptual or calculation errors.
In the given function , is a constant. The graph of function in the xy-plane, where , is translated units down and units to the right to produce the graph of . Which equation defines function ?
Choice B is correct. It's given that the graph of is produced by translating the graph of units down and units to the right in the xy-plane. Therefore, function can be defined by an equation in the form . Function is defined by the equation , where is a constant. Substituting for in the equation yields . Substituting for in the equation yields , or . Therefore, the equation that defines function is .
Choice A is incorrect. This equation defines a function whose graph is produced by translating the graph of units down and units to the left, not units down and units to the right.
Choice C is incorrect. This equation defines a function whose graph is produced by translating the graph of units to the left, not units down and units to the right.
Choice D is incorrect. This equation defines a function whose graph is produced by translating the graph of units to the right, not units down and units to the right.
The given equation relates the positive numbers , , and . Which equation correctly expresses in terms of and ?
Choice A is correct. Adding to each side of the given equation yields .
Choice B is incorrect. This equation is equivalent to , not .
Choice C is incorrect. This equation is equivalent to , not .
Choice D is incorrect. This equation is equivalent to , not .
If , which of the following is a possible value of x ?
1
Choice D is correct. If , then taking the square root of each side of the equation gives
or
. Solving these equations for x gives
or
. Of these,
is the only solution given as a choice.
Choice A is incorrect and may result from solving the equation and making a sign error. Choice B is incorrect and may result from solving the equation
. Choice C is incorrect and may result from finding a possible value of
.
The function is defined by the given equation. The equation can be rewritten as , where is a constant. Which of the following is closest to the value of ?
Choice A is correct. The equation can be rewritten as , which is equivalent to , or approximately . Since it's given that can be rewritten as , where is a constant, it follows that is approximately equal to . Therefore, is approximately equal to . It follows that the value of is approximately equal to . Of the given choices, is closest to the value of .
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
What is the minimum value of the function f defined by ?
2
4
Choice A is correct. The given quadratic function f is in vertex form, , where
is the vertex of the graph of
in the xy-plane. Therefore, the vertex of the graph of
is
. In addition, the y-coordinate of the vertex represents either the minimum or maximum value of a quadratic function, depending on whether the graph of the function opens upward or downward. Since the leading coefficient of f (the coefficient of the term
) is 1, which is positive, the graph of
opens upward. It follows that at
, the minimum value of the function f is
.
Choice B is incorrect and may result from making a sign error and from using the x-coordinate of the vertex. Choice C is incorrect and may result from using the x-coordinate of the vertex. Choice D is incorrect and may result from making a sign error.
A function estimates that there were animals in a population in . Each year from to , the function estimates that the number of animals in this population increased by of the number of animals in the population the previous year. Which equation defines this function, where is the estimated number of animals in the population years after ?
Choice C is correct. It's given that a function estimates that there were animals in a population in and that each year from to , the number of animals in this population increased by of the number of animals in the population the previous year. It follows that this situation can be represented by the function , where is the estimated number of animals in the population years after , is the estimated number of animals in the population in , and each year the estimated number of animals increased by . Substituting for and for in this function yields , or .
Choice A is incorrect. This function represents a population in which each year the number of animals increased by , not , of the number of animals in the population the previous year.
Choice B is incorrect. This function represents a population in which each year the number of animals increased by , not , of the number of animals in the population the previous year.
Choice D is incorrect. This function represents a population in which each year the number of animals decreased, rather than increased, by of the number of animals in the population the previous year.
What is the negative solution to the given equation?
The correct answer is . Dividing both sides of the given equation by yields . Taking the square root of both sides of this equation yields the solutions and . Therefore, the negative solution to the given equation is .
In the equation above, a and c are positive constants. How many times does the graph of the equation above intersect the graph of the equation in the xy-plane?
Zero
One
Two
More than two
Choice C is correct. It is given that the constants a and c are both positive; therefore, the graph of the given quadratic equation is a parabola that opens up with a vertex on the y-axis at a point below the x-axis. The graph of the second equation provided is a horizontal line that lies above the x-axis. A horizontal line above the x-axis will intersect a parabola that opens up and has a vertex below the x-axis in exactly two points.
Choices A, B, and D are incorrect and are the result of not understanding the relationships of the graphs of the two equations given. Choice A is incorrect because the two graphs intersect. Choice B is incorrect because in order for there to be only one intersection point, the horizontal line would have to intersect the parabola at the vertex, but the vertex is below the x-axis and the line is above the x-axis. Choice D is incorrect because a line cannot intersect a parabola in more than two points.
Let the function p be defined as , where c is a constant. If
, what is the value of
?
10.00
10.25
10.75
11.00
Choice D is correct. The value of p(12) depends on the value of the constant c, so the value of c must first be determined. It is given that p(c) = 10. Based on the definition of p, it follows that:
This means that for all values of x. Therefore:
Choice A is incorrect. It is the value of p(8), not p(12). Choices B and C are incorrect. If one of these values were correct, then x = 12 and the selected value of p(12) could be substituted into the equation to solve for c. However, the values of c that result from choices B and C each result in p(c) < 10.
The given equation relates the positive numbers , , and . Which equation correctly expresses in terms of and ?
Choice B is correct. To express in terms of and , the given equation can be rewritten such that is isolated on one side of the equation. Since it’s given that is a positive number, is not equal to zero. Therefore, dividing both sides of the given equation by yields the equivalent equation , or .
Choice A is incorrect. This equation is equivalent to .
Choice C is incorrect. This equation is equivalent to .
Choice D is incorrect. This equation is equivalent to .
What value of x satisfies the equation above?
The correct answer is 117. Squaring both sides of the given equation gives , or
. Subtracting 4 from both sides of this equation gives
.
What is the sum of the solutions to the given equation?
Choice D is correct. By the definition of absolute value, if , then or . Subtracting from both sides of the equation yields . Dividing both sides of this equation by yields . Subtracting from both sides of the equation yields . Dividing both sides of this equation by yields . Therefore, the solutions to the given equation are and , and it follows that the sum of the solutions to the given equation is , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is a solution, not the sum of the solutions, to the given equation.
Choice C is incorrect and may result from conceptual or calculation errors.
The function is defined by . The value of is , where is a constant. What is the sum of all possible values of ?
The correct answer is . The value of is the value of when , where is a constant. Substituting for in the given equation yields , which is equivalent to . It’s given that the value of is . Substituting for in the equation yields . Since the product of the three factors on the right-hand side of this equation is equal to , at least one of these three factors must be equal to . Therefore, the possible values of can be found by setting each factor equal to . Setting the first factor equal to yields . Adding to both sides of this equation yields . Therefore, is one possible value of . Setting the second factor equal to yields . Adding to both sides of this equation yields . Therefore, is a second possible value of . Setting the third factor equal to yields . Taking the square root of both sides of this equation yields . Adding to both sides of this equation yields . Therefore, is a third possible value of . Adding the three possible values of yields , or . Therefore, the sum of all possible values of is .
The function f is defined above. What is the value of ?
250
500
750
2,000
Choice C is correct. Adding the like terms x and yields the equation
. Substituting 20 for x yields
. The product
is equal to 25, and the difference
is equal to 30. Substituting these values in the given equation gives
, and multiplying 25 by 30 results in
.
Choices A, B, and D are incorrect and may result from conceptual or computational errors when finding the value of .
Which expression is equivalent to ?
Choice B is correct. The given expression may be rewritten as . Since the first two terms of this expression have a common factor of and the last two terms of this expression have a common factor of , this expression may be rewritten as , or . Since each term of this expression has a common factor of , it may be rewritten as .
Alternate approach: An expression of the form , where and are constants, can be factored if there are two values that add to give and multiply to give . In the given expression, and . The values of and add to give and multiply to give , so the expression can be factored as .
Choice A is incorrect. This expression is equivalent to , not .
Choice C is incorrect. This expression is equivalent to , not .
Choice D is incorrect. This expression is equivalent to , not .
There were no jackrabbits in Australia before 1788 when 24 jackrabbits were introduced. By 1920 the population of jackrabbits had reached 10 billion. If the population had grown exponentially, this would correspond to a 16.2% increase, on average, in the population each year. Which of the following functions best models the population of jackrabbits t years after 1788?
Choice C is correct. This exponential growth model can be written in the form , where
is the population t years after 1788, A is the initial population, and r is the yearly growth rate, expressed as a decimal. Since there were 24 jackrabbits in Australia in 1788,
. Since the number of jackrabbits increased by an average of 16.2% each year,
. Therefore, the equation that best models this situation is
.
Choices A, B, and D are incorrect and may result from misinterpreting the form of an exponential growth model.
Which of the following expressions is equivalent to the sum of and
?
Choice D is correct. Grouping like terms, the given expressions can be rewritten as . This can be rewritten as
.
Choice A is incorrect and may result from adding the two sets of unlike terms, and
as well as
and
, and then adding the respective exponents. Choice B is incorrect and may result from adding the unlike terms
and
as if they were
and
and adding the unlike terms
and
as if they were
and
. Choice C is incorrect and may result from errors when combining like terms.
Which expression is equivalent to ?
Choice C is correct. Each term in the given expression, , has a common factor of . Therefore, the expression can be rewritten as , or . Thus, the expression is equivalent to the given expression.
Choice A is incorrect. This expression is equivalent to , not .
Choice B is incorrect. This expression is equivalent to , not .
Choice D is incorrect. This expression is equivalent to , not .
A right rectangular prism has a height of inches. The length of the prism's base is inches, which is inches more than the width of the prism's base. Which function gives the volume of the prism, in cubic inches, in terms of the length of the prism's base?
Choice D is correct. The volume of a right rectangular prism can be represented by a function that gives the volume of the prism, in cubic inches, in terms of the length of the prism's base. The volume of a right rectangular prism is equal to the area of its base times its height. It's given that the length of the prism's base is inches, which is inches more than the width of the prism's base. This means that the width of the prism's base is inches. It follows that the area of the prism's base, in square inches, is and the volume, in cubic inches, of the prism is . Thus, the function that gives the volume of this right rectangular prism, in cubic inches, in terms of the length of the prism's base, , is .
Choice A is incorrect. This function would give the volume of the prism if the height were inches more than the length of its base and the width of the base were inches more than its length.
Choice B is incorrect. This function would give the volume of the prism if the height were inches more than the length of its base.
Choice C is incorrect. This function would give the volume of the prism if the width of the base were inches more than its length, rather than the length of the base being inches more than its width.
The solution to the given system of equations is . What is the value of ?
The correct answer is . Adding to both sides of the second equation in the given system yields . Substituting for in the first equation in the given system yields . Subtracting from both sides of this equation yields . Factoring the left-hand side of this equation yields , or . Taking the square root of both sides of this equation yields . Subtracting from both sides of this equation yields . Therefore, the value of is .